I 


i 


CONTRIBUTIONS     TO 

THE   FOUNDING  OF  THE  THEORY  OF 

TRANSFINITE  NUMBERS 


/  Court  Series  of  Classics  of  Science  and 
Thilosophy,  ,?(o.  i 

TRIBUTIONS    TO 

INDING  OF  THE  THEORY  OF 

Ik        SFINITE   NUMBERS 


BY 

GEORG   CANTOR 


TRANSLATED,  AND    PROVIDED    WITH    AN    INTRODUCTION 
AND    NOTES,   BY 

PHILIP   E.  B.  JOURDAIN 

M.  A.  (Cantab.) 


CHICAGO   AND    LONDON 
PEN  COURT  PUBLISHING  COMPANY 
1915 


G2 


Copyright  in  Great  Britain  under  the  Act  of  igii 


PREFACE 

This  volume  contains  a  translation  of  the  two  very 
iniportant  memoirs  of  Georg  Cantor  on  transfinite 
numbers  which  appeared  in  the  Mathematische 
Annalen  for  1895  ^^'^^  1897*  under  the  title: 
''Beitrage  zur  Begriindung  der  transfmiten  Mengen- 
lehre."  It  seems  to  me  that,  since  these  memoirs 
are  chiefly  occupied  with  the  investigation  of  the 
various  transfinite  cardinal  and  ordinal  numbers  and 
not  with  investigations  belonging  to  what  is  usually 
described  as  *'the  theory  of  aggregates"  or  "the 
theory  of  sets  "  {Mengenlehre^  theorie  des  ensembles), 
— the  elements  of  the  sets  being  real  or  complex 
numbers  which  are  imaged  as  geometrical  "  points  " 
in  space  of  one  or  more  dimensions, — the  title  given 
to  them  in  this  translation  is  more  suitable. 

These  memoirs  are  the  final  and  logically  purified 
statement  of  many  of  the  most  important  results  of 
the  long  series  of  memoirs  begun  by  Cantor  in  1870. 
It  is,  I  think,  necessary,  if  we  are  to  appreciate  the 
full  import  of  Cantor's  work  on  transfinite  numbers, 
■'  ->  ^ave  thought  through  and  to  bear  in  mind  Cantor's 

'  researches  on  the  theory  of  point-aggregates. 

s   in  these   researches    that    the   need   for   the 

^ol,  xlvi,  1895,  pp.  481-512  ;  vol.  xlix,  1897,  pp.  207-246. 

334640 


vi  PREFACE 

transfinite  numbers  first  showed  itself,  and  it  is  only 
by  the  study  of  these  researches  that  the  majority 
of  us  can  annihilate  the  feeling  of  arbitrariness  and 
even  insecurity  about  the  introduction  of  these 
numbers.  Furthermore,  it  is  also  necessary  to  trace 
backwards,  especially  through  Weierstrass,  the 
course  of  those  researches  which  led  to  Cantor's 
work.  I  have,  then,  prefixed  an  Introduction  tracing 
the  growth  of  parts  of  the  theory  of  functions  during 
the  nineteenth  century,  and  dealing,  in  some  detail, 
with  the  fundamental  work  of  Weierstrass  and  others. 
and  with  the  work  of  Cantor  from  1870  to  1895. 
Some  notes  at  the  end  contain  a  short  account  of  the 
developments  of  the  theory  of  transfinite  numbers 
since  1897.  I^  these  notes  and  in  the  Introduction 
I  have  been  greatly  helped  by  the  information  that 
Professor  Cantor  gave  me  in  the  course  of  a  long 
correspondence  on  the  theory  of  aggregates  which 
we  carried  on  many  years  ago. 

The  philosophical  revolution  brought  at  out  by 
Cantor's  work  was  even  greater,  perhaps,  than  the 
mathematical  one.  With  few  exceptions,  mathe- 
maticians joyfully  accepted,  built  upon,  scrutinized, 
and  perfected  the  foundations  of  Cantor's  u  idying 
theory;  but  very  many  philosophers  combaied  it. 
This  seems  to  have  been  because  very  few  under- 
stood it.  I  hope  that  this  book  may  help  to  make 
the  subject  better  known  to  both  philosophcs  and 
mathematicians. 

The  three  men  whose  influence  on  moder;i  pure 
mathematics — and  indirectly  modern  logic  and  the 


PREFACE  vii 

philosophy  which  abuts  on  it — is  most  marked  are 
Karl  Weierstrass,  Richard  Dedekind,  and  Georg 
Cantor.  A  great  part  of  Dedekind's  work  has  de- 
veloped along  a  direction  parallel  to  the  work  of 
Cantor,  and  it  is  instructive  to  compare  with  Cantor's 
work  Dedekind's  Stetigkeit  unci  irrationale  Zahlen 
ar^d  Was  sind  und  was  sollen  die  Zahlen  P,  of  which 
excellent  English  translations  have  been  issued  by 
the  publishers  of  the  present  book.  * 

There  is  a  French  translation  f  of  these  memoirs  of 
Cantor's,  but  there  is  no  English  translation  of  them. 
For  kind  permission  to  make  the  translation,  1 
am  indebted  to  Messrs  B.  G.  Teubner  of  Leipzig 
and  Berlin,  the  publishers  of  the  Matheinatische 
Annalen. 

PHILIP  E.  B.  JOURDALN. 

*  Essays  on  the  Theory  of  Numbers  (I,  Continuity  and  Irrational 
Numbers;  II,  The  Nature  arid  Meaning- of  Numbers),  translated  by 
W.  W.  Reman,  Chicago,  190T.  I  shall  refer  to  this  as  Essays  on 
Number. 

t  By  F.  Marotte,  Sur  les  fondements  de  la  theorie  des  ensembles 
trans/mis,  Paris,  1899. 


TABLE   OF   CONTENTS 


I'AGE 


Preface v 

Table  of  Contents ix 

Introduction i 

Contributions  to  the  Founding  of  the  Theory 
OF  Transftnite  Numbers — 

Article  I.  (1895) 85 

Article  II.  (1897) -137 

Notes 202 

Index 209 


CONTRIBUTIONS  TO  THE 

FOUNDING  OF   THE   THEORY 

OF  TRANSFINITE  NUMBERS 


INTRODUCTION 

I 
If  it  is  safe  to  trace  back  to  any  single  man  the 
origin  of  those  conceptions  with  which  pure  mathe- 
matical analysis  has  been  chiefly  occupied  during 
the  nineteenth  century  and  up  to  the  present  time, 
we  must,  I  think,  trace  it  back  to  Jean  Baptiste 
Joseph  Fourier  (1768- 1830).  Fourier  was  first  and 
foremost  a  physicist,  and  he  expressed  very  defin- 
itely his  view  that  mathematics  only  justifies  itself 
.by  the  help  it  gives  towards  the  solution  of  physical 
problems,  and  yet  the  light  that  was  thrown  on  the 
general  conception  of  a  function  and  its  ''con- 
tinuity," of  the  ''convergence"  of  infinite  series, 
and  of  an  integral,  first  began  to  shine  as  a  result 
of  Fourier's  original  and  bold  treatment  of  the 
problems  of  the  conduction  of  heat.  This  it  was 
that  gave  the  impetus  to  the  formation  and  develop- 
ment of  the  theories  of  functions.  The  broad- 
minded     physicist    will    approve    of    this    refining 


1  INTRODUCTION 

development  of  the  mathematical  methods  which 
arise  from  physical  conceptions  when  he  reflects 
that  mathematics  is  a  wonderfully  powerful  and 
economically  contrived  means  of  dealing  logically 
and  conveniently  with  an  immense  complex  of  data, 
and  that  we  cannot  be  sure  of  the  logical  soundness 
of  our  methods  and  results  until  we  make  every- 
thing about  them  quite  definite.  The  pure  mathe- 
matician knows  that  pure  mathematics  has  an  end 
in  itself  which  is  more  allied  with  philosophy.  But 
we  have  not  to  justify  pure  mathematics  here  :  we 
have  only  to  point  out  its  origin  in  physical  con- 
ceptions. But  we  have  also  pointed  out  that 
physics  can  justify  even  the  most  modern  develop- 
ments of  pure  mathematics. 

II 

During    the    nineteenth    century,   the    two    great 
branches  of  the  theory  of  functions  developed  and 
gradually  separated.      The    rigorous    foundation    of 
the    results    of    P^ourier    on    trigonometrical    series, 
which  was  given  by  Dirichlet,  brought   forward  as 
subjects  of  investigation  the  general  conception  of  a 
(one-valued)  function  of  a  real  variable  and  the  (in 
particular,  trigonometrical)  development  of  functions. 
On    th^  other  hand,   Cauchy  was  gradually  led  to 
recognize  the  importance  of  what  was  subsequently 
seen  to  be  the  more  special  conception  of  function  of 
a  complex  variable  ;  and,  to  a  great  extent  independ- 
ently of  Cauchy,  Weierstrass  built  up  his  theory  of 
analytic  functions  of  complex  variables. 


INTRODUCTION  3 

These  tendencies  of  both  Cauchy  and  Dirichlet 
combined  to  influence  Riemann  ;  his  work  on  the 
theory  of  functions  of  a  conaplex  variable  carried  on 
and  greatly  developed  the  work  of  Cauchy,  while 
the  intention  of  his  '' Habilitationsschrift  "  of  1854 
was  to  generalize  as  far  as  possible  Dirichlet's  partial 
solution  of  the  problem  of  the  development  of  a 
function  of  a  real  variable  in  a  trigonometrical 
series. 

Both  these  sides  of  Riemann's  activity  left  a  deep 
impression  on  Hankel.  In  a  memoir  of  1870, 
Hankel  attempted  to  exhibit  the  theory  of  functions 
of  a  real  variable  as  leading,  of  necessity,  to  the 
restrictions  and  extensions  from  which  we  start  in 
Riemann's  theory  of  functions  of  a  complex  variable  ; 
and  yet  Hankel's  researches  entitle  him  to  be  called 
the  founder  of  the  independent  theory  of  functions 
of  a  real  variable.  At  about  the  same  time,  Heine 
initiated,  under  the  direct  influence  of  Riemann's 
'*  Habilitationsschrift,"  a  new  series  of  investigations 
on  trigonometrical  series. 

Finally,  soon  after  this,  we  find  Georg  Cantor 
both  studying  Hankel's  me.moir  and  applying  to 
theorems  on  the  uniqueness  of  trigonometrical  de- 
velopments those  conceptions  of  his  on  irrational 
numbers  and  the  'derivatives"  of  point-aggregates 
or  number-aggregates  which  developed  from  the 
rigorous  treatment  of  such  fundamental  questions 
given  by  Weierstrass  at  Berlin  in  the  introduction  to 
his  lectures  on  analytic  functions.  The  theory  of 
point-aggregates  soon  became  an  independent  theory 


4  INTRODUCTION 

of  great  importance,  and  finally,  in  1882,  Cantor's 
^'transfinite  numbers"  were  defined  independently 
of  the  aggregates  in  connexion  with  which  they  first 
appeared  in  mathematics. 

Ill 

The  investigations  *  of  the  eighteenth  century  on 
the  problem  of  vibrating  cords  led  to  a  controversy 
for  the  following  reasons.  D'Alembert  maintained 
that  the  arbitrary  functions  in  his  general  integral 
of  the  partial  differential  equation  to  which  this 
problem  led  were  restricted  to  have  certain  pro- 
perties which  assimilate  them  to  the  analytically 
representable  functions  then  known,  and  which  would 
prevent  their  course  being  completely  arbitrary  at 
every  point.  Euler,  on  the  other  hand,  argued  for 
the  admission  of  certain  of  these  "arbitrary" 
functions  into  analysis.  Then  Daniel  BernouUi 
produced  a  solution  in  the  form  of  an  infnite 
trigonometrical  series,  and  claimed,  on  certain 
physical  grounds,  that  this  solution  was  as  general 
as  d'Alembert's.  As  Euler  pointed  out,  this  was  so 
only  if  any  arbitrary  f  function  ^(;ir)  were  develops 
able  in  a  series  of  the  form 

*  Cf.  the  references  given  in  my  papers  in  the  Archiv  der  Matkematfk 
liud  Physik,  3rd  series,  vol.  x,  1906,  pp.  255-256,  and  his,  vol.  i, 
1914,  pp.  670-677.  Much  of  this  Introduction  is  taken  frorr.  my 
account  of  "The  Development  of  the  Theory  of  Transfinite  Number-i" 
in  the  above-mentioned  Archiv,  3rd  series,  vol,  x,  pp.  254-2S1  ; 
vol.  xiv,  1909,  pp.  289-311;  vol.  xvi,  1910,  pp.  21-43;  vol.  j  tii, 
1913,  pp.  1-21. 

t  The  arbitrary  functions  chiefly  considered  in  this  connexio  i  by 
Euler  were  what  he  called  "discontinuous"  functions.  This  ^/ord 
does  not  mean  what  we  now  mean  (after  Cauchy)  by  it.  Cj\  my  paper 
in  his,  vol.  i,  191 4,  pp.  661-703. 


INTRODUCTION 


vkx 


That  this  was,  indeed,  the  case,  even  when  0(;f) 
is  not  necessarily  developable  in  a  power-series,  was 
first  shown  by  Fourier,  who  was  led  to  study  the 
same  mathematical  problem  as  the  above  one  b}' 
his  researches,  the  first  of  which  were  communicated 
to  the  French  Academy  in  1807,  on  the  conduction 
of  heat.  To  Fourier  is  due  also  the  determination 
of  the  coefficients  in  trigonometric  series, 

0(;ir)  =  J/^o  +  ^i  cos^'H-<^2  ^os  2;f+  .  .  . 
-\-a^?A\-\  x-\-a^^^\\\2x-\-  .  .  ., 
in  the  form 

d^=-  I (l>{a)  cos  vada,      «,.  =  —  I  0(a)  sin  vada. 
it]  ttJ 

-n  -7T 

This  determination  was  probably  independent  of 
Euler's  prior  determination  and  Lagrange's  analog- 
ous determination  of  the  coefficients  of  a  Jinzte 
trigonometrical  series.  Fourier  also  gave  a  geo- 
metrical proof  of  the  convergence  of  his  series, 
which,  though  not  formally  exact,  contained  the 
germ  of  Dirichlet's  proof. 

To  Peter  Gustav  Lejeune-Dirichlet  (1805-1859) 
is  due  the  first  exact  treatment  of  Fourier's  series.* 
He  expressed  the  sum  of  the  first  n  terms  of  the 
series  by  a  definite  integral,   and    proved  that   the 

"'•  "Sur  la  convergence  des  series  trigonometriques  qui  servent  a 
representer  une  fonction  arbitraire  entre  des  liniites  donnees,"y<?«;-«. 
fur  Math.,  vol.  iv,  1829,  pp.  157-169;  Ges.  Werke^  vol.  i, 
pp.    1 17-132, 


6  INTRODUCTION 

limit,  when  n  increases  indefinitely,  of  this  integral 
is  the  function  which  is  to  be  represented  by  the 
trigonometrical  series,  provided  that  the  function 
satisfies  certain  conditions.  These  conditions  were 
somewhat  lightened  by  Lipschitz  in  1864. 

Thus,  Fourier's  work  led  to  the  contem.plation 
and  exact  treatment  of  certain  functions  which 
were  totally  different  in  behaviour  from  algebraic 
functions.  These  last  functions  were,  before  him, 
tacitly  considered  to  be  the  type  of  all  functions  that 
can  occur  in  analysis.  Henceforth  it  was  part  of 
the  business  of  analysis  to  investigate  such  non- 
algebraoid  functions. 

In  the  first  few  decades  of  the  nineteenth  century 
there  grew  up  a  theory  of  more  special  functions  of 
an  imaginary  or  complex  variable.  This  theory  was 
known,  in  part  at  least,  to  Carl  Friedrich  Gauss 
(1777-1855),  but  he  did  not  publish  his  results,  and 
so  the  theory  is  due  to  Augustin  Louis  Cauchy 
(1789- 1 857).*  Cauchy  was  less  far-sighted  and 
penetrating  than  Gauss,  the  theory  developed 
slowly,  and  only  gradually  were  Cauchy's  prejudices 
against  ' '  imaginaries "  overcome.  Through  the 
years  from  18 14  to  1846  we  can  trace,  first,  the 
strong  influence  on  Cauchy's  conceptions  of  Fourier's 
ideas,  then  the  quickly  increasing  unsusceptibility  to 
the  ideas  of  others,  coupled  with  the  extraordinarily 
prohfic  nature  of  this  narrow-minded  genius.  Cauchy 
appeared  to  take  pride  in  the  production  of  memoirs 

*   Cf.  Jourdain,  "The  Theory  of  Functions  with  Cauchy  and  Gauss," 
Bibl.'Maih.  (3),  vol.  vi,  1905,  pp.  190-207. 


INTRODUCTION  7 

at  each  weekly  meeting  of  the  French  Academy,  and 
it  was  partly,  perhaps,  due  to  this  circumstance  that 
his  works  are  of  very  unequal  importance.  Besides 
that,  he  did  not  seem  to  perceive  even  approximately 
the  immense  importance  of  the  theory  of  functions 
of  a  complex  variable  which  he  did  so  much  to 
create.  This  task  remained  for  Puiseux,  Briot  and 
Bouquet,  and  others,  and  was  advanced  in  the 
most  striking  manner  by  Georg  Friedrich  Bernhard 
Riemann  (1826-1866). 

Riemann  may  have  owed  to  his  teacher  Dirichlet 
his  bent  both  towards  the  theory  of  potential — 
which  was  the  chief  instrument  in  his  classical 
development  (185  i)  of  the  theory  of  functions  of  a 
complex  variable — and  that  of  trigonometrical  series. 
By  a  memoir  on  the  representability  of  a  function 
by 'a  trigonometrical  series,  which  was  read  in  1854 
but  only  published  after  his  death,  he  not  only  laid 
the  foundations  for  all  modern  investigations  into  the 
theory  of  these  series,  but  inspired  Hermann  Hankel 
(1839- 1 873)  to  the  method  of  researches  from  which: 
we  can  date  the  theory  of  functions  of  a  real  variable 
as  an  independent  science.  The  motive  of  Hankel's 
research  was  provided  by  reflexion  on  the  founda- 
tions of  Riemann's  theory  of  functions  of  a  complex 
variable.  It  was  Hankel's  object  to  show  how  the 
needs  of  mathematics  compel  us  to  go  beyond  the 
most  general  conception  of  a  function,  which  was 
implicitly  formulated  by  Dirichlet,  to  introduce  the 
complex  variable,  and  finally  to  reach  that  con- 
ception from  which  Riemann  started  in  his  inaugural 


8  INTRODUCTION 

dissertation.  For  this  purpose  Hankel  began  his 
'' Untersuchungen  liber  die  unendlich  oft  oscilli- 
renden  und  unstetigen  Functionen  ;  ein  Beitrag  zur 
Feststelkmg  des  Begriffes  der  Function  liberhaupt  " 
of  1870  by  a  thorough  examination  of  the  various 
possibihties  contained  in  Dirichlet's  conception. 

Riemann,  in  his  memoir  of  1854,  started  from 
the  general  problem  of  which  Dirichlet  had  only 
solved  a  particular  case  :  If  a  function  is  developable 
in  a  trigonometrical  series,  what  results  about  the 
variation  of  the  value  of  the  function  (that  is  to  say, 
what  is  the  most  general  way  in  which  it  can  become 
discontinuous  and  have  maxima  and  minima)  when 
the  argument  varies  continuously  ?  The  argument 
is  a  real  variable,  for  Fourier's  series,  as  Fourier  had 
already  noticed,  may  converge  for  real  x'<^  alone. 
This  question  was  not  completely  answered,  and, 
perhaps  in  consequence  of  this,  the  work  was  not 
published  in  Riemann's  lifetime  ;  but  fortunately 
that  part  of  it  which  concerns  us  more  particularly, 
and  which  seems  to  fill,  and  more  than  fill,  the  place 
of  Dirichlet's  contemplated  revision  of  the  principles 
of  the  infinitesimal  calculus,  has  the  finality  obtained 
by  the  giving  of  the  necessary  and  sufficient  condi- 
tions for  the  integrability  of  a  function  f{x),  which 
was  a  necessary  preliminary  to  Riemann's  investiga- 
tion. Thus,  Riemann  was  led  to  give  the  process 
of  integration  a  far  wider  meaning  than  that 
contemplated  by  Cauchy  or  even  Dirichlet,  and 
Riemann  constructed  an  integrable  function  which 
becomes  discontinuous  an  infinity  of  times  between 


INTRODUCTION  9 

any  two  limits,  as  close  together  as  wished,  of  the 
independent  variable,  in  the  following  manner  : — If, 
where  ;r  is  a  real  variable,  (x)  denotes  the  (positive  or 
negative)  excess  of  x  over  the  nearest  integer,  or 
zero  if  x  is  midway  between  two  integers,  {x)  is 
a  one-valued  function  of  x  with  discontinuities  at 
the  points  ^=;^  +  i,  where  n  is  an  integer  (positive, 
negative,  or  zero),  and  with  \  and  —  \  for  upper  and 
lower  limits  respectively.  Further,  (la-),  where  v  is 
an  integer,  is  discontinuous  at  the  points  vx=n-\-\ 

ox  x—-{7i-\-\).      Consequentl}-,  the  series 

l'=l  ^ 

where  the  factor  \\v^  is  added  to  ensure  convergence 
for  all  values  of  ;r,  may  be  supposed  to  be  discon- 
tinuous for  all  values  of  x  of  the  form  x=p\2n^ 
where/  is  an  odd  integer,  relatively  prime  to  ;^.  It 
was  this  method  that  was,  in  a  certain  respect, 
generalized  by  Hankel  in  1870.  In  Riemann's 
example  appeared  an  analytical  expression — and 
therefore  a  ''  function  "  in  Euler's  sense — which,  on 
account  of  its  manifold  singularities,  allowed  of  no 
such  general  properties  as  Riemann's  "  functions  of 
a  complex  variable,"  and  Hankel  gave  a  method, 
whose  principles  were  suggested  by  this  example,  of 
forming  analytical  expressions  with  singularities  at 
every  rational  point.  He  was  thus  led  to  state,  with 
some  reserve,  that  every  ''function"  in  Dirichlet's 
sense  is  also  a  "  function  "  in  Euler's  sense. 

The   greatest  influence  on  Georg  Cantor  seems, 


10  INTRODUCTION 

however,  not  to  have  been  that  exercised  by 
Riemann,  Hankel,  and  their  successors — though 
the  work  of  these  men  is  closely  connected  with 
some  parts  of  Cantor's  work, — but  by  Weierstrass, 
a  contemporary  of  Riemann's,  who  attacked  many 
of  the  same  problems  in  the  theory  of  analytic 
functions  of  complex  variables  by  very  different  and 
more  rigorous  methods. 

IV 

Karl  Weierstrass  (i8i 5-1897)  has  explained,  in 
his  address  delivered  on  the  occasion  of  his  entry 
into  the  Berlin  Academy  in  1857,  that,  from  the 
time  (the  winter  of  18  39- 1840)  when,  under  his 
teacher  Gudermann,  he  made  his  first  acquaintance 
with  the  theory  of  elliptic  functions,  he  was  power- 
fully attracted  by  this  branch  of  analysis.  "  Now, 
Abel,  who  was  accustomed  to  take  the  highest 
standpoint  in  any  part  of  mathematics,  established 
a  theorem  which  comprises  all  those  transcendents 
which  arise  from  the  integration  of  algebraic  differ- 
entials, and  has  the  same  signification  for  these  as 
Euler's  integral  has  for  elliptic  functions  .  .  .  ;  and 
Jacobi  succeeded  in  demonstrating  the  existence  of 
periodic  functions  of  many  arguments ,  whose  funda- 
mental properties  are  established  in  Abel's  theorem, 
and  by  means  of  which  the  true  meaning  and  real 
essence  of  this  theorem  could  be  judged.  Actually 
to  represent,  and  to  investigate  the  properties  of 
these  magnitudes  of  a  totally  new  kind,  of  which 
analysis  has  las  yet  no  example,  I  regarded  as  one 


INTRODUCTION  ii 

of  the  principal  problems  of  mathematics,  and,  as 
soon  as  I  clearly  recognized  the  meaning  and  sig- 
nificance of  this  problem,  resolved  to  devote  myself 
to  it.  Of  course  it  would  have  been  foolish  even 
to  think  of  the  solution  of  such  a  problem  without 
having  prepared  myself  by  a  thorough  study  of  the 
means  and  by  busying  myself  with  less  difficult 
problems. " 

VV^ith  the  ends  stated  here  of  Weierstrass's  work 
we  are  now  concerned  only  incidentally  :  it  is  the 
means — the  ' '  thorough  study  "  of  which  he  spoke — 
which  has  had  a  decisive  influence  on  our  subject  in 
common  with  the  theory  of  functions.  We  will, 
then,  pass  over  his  early  work — which  was  only 
published  in  1894 — o^^  the  theory  of  analytic 
functions,  his  later  work  on  the  same  subject,  and 
his  theory  of  the  Abelian  functions,  and  examine 
his  immensely  important  work  on  the  foundations 
of  arithmetic,  to  which  he  was  led  by  the  needs  of 
a  rigorous  theory  of  analytic  functions. 

We  have  spoken  as  if  the  ultimate  aim  of  Weier- 
strass's work  was  the  investigation  of  Abelian 
functions.  But  another  and  more  philosophical 
view  was  expressed  in  his  introduction  to  a  course 
of  lectures  delivered  in  the  summer  of  1886  and 
preserved  by  Gosta  Mittag-Leffler  *  :  '*  In  order  to 
penetrate  into  mathematical  science  it  is  indispens- 
able that  we  should  occupy  ourselves  with  individual 


*  "  Sur  les  fondements  arithmetiques  cle  la  theorie  des  fonctions 
d'apres  Weierstrass,"  Congees  des  Alath^ma/iques  ct  Shnk/iohn, 
1909,  p.  10. 


12  INTRODUCTION 

problems  which  show  us  its  extent  and  constitution. 
But  the  final  object  which  we  must  always  keep  in 
sight  is  the  attainment  of  a  sound  judgment  on  the 
foundations  of  science." 

In  1859,  Weierstrass  began  his  lectures  on  the 
theory  of  analytic  functions  at  the  University  of 
Berlin.  The  importance  of  this,  from  our  present 
point  of  view,  lies  in  the  fact  that  he  was  naturally 
obliged  to  pay  special  attention  to  the  systematic 
treatment  of  the  theory,  and  consequently,  to 
scrutinize  its  foundations. 

In  the  first  place,  one  of  the  characteristics  of 
Weierstrass's  theory  of  functions  is  the  abolition  of 
the  method  of  complex  integration  of  Cauchy  and 
Gauss  which  was  used  by  Riemann  ;  and,  in  a 
letter  to  H.  A.  Schwarz  of  October  3,  1875, 
Weierstrass  stated  his  belief  that,  in  a  systematic 
foundation,  it  is  better  to  dispense  with  integration, 
as  follows  :— 

''.  .  .  The  more  I  meditate  upon  the  principles 
of  the  theory  of  functions, — and  I  do  this  incessantly, 
— the  firmer  becomes  my  conviction  that  this  theory 
must  be  built  up  on  the  foundation  of  algebraic 
truths,  and  therefore  that  it  is  not  the  right  way  to 
proceed  conversely  and  make  use  of  the  trans- 
cendental (to  express  myself  briefly)  for  the  establish- 
ment of  simple  and  fundamental  algebraic  theorems  ; 
however  attractive  may  be,  for  example,  the  con- 
siderations by  which  Riemann  discovered  so  many 
of  the  most  important  properties  of  algebraic 
functions,      That  to  the  discoverer,  qud  discoverer. 


INTRODUCTION  13 

every  route  is  permissible,  is,  of  course,  self-evident  ; 
I  am  only  thinking  of  the  systematic  establishment 
of  the  theory. " 

In  the  second  place,  and  what  is  far  more  im- 
portant than  the  question  of  integration,  the 
systematic  treatment,  ab  initio^  of  the  theory  of 
analytic  functions  led  Weierstrass  to  profound  in- 
vestigations in  the  principles  of  arithmetic,  and  the 
great  result  of  these  investigations — his  theory  of 
irrational  numbers — has  a  significance  for  all  mathe- 
matics which  can  hardly  be  overrated,  and  our 
present  subject  may  truly  be  said  to  be  almost 
'wholly  due  to  this  theory  and  its  development  by 
Cantor. 

In  the  theory  of  analytic  functions  we  often  have 
to  use  the  theorem  that,  if  we  are  given  an  infinity 
of  points  of  the  complex  plane  in  any  bounded 
region  of  this  plane,  there  is  at  least  one  point  of 
the  domain  such  that  there  is  an  infinity  of  the 
given  points  in  each  and  every  neighbourhood  round 
it  and  including  it.  Mathematicians  used  to  express 
this  by  some  such  rather  obscure  phrase  as  :  * '  There 
is  a  point  near  which  some  of  the  given  points  are 
lriHhil;ely  near  to  one  another."  If  we  apply,  for  the 
proof  of  this,  the  method  which  seems  naturally  to 
suggest  itself,  and  which  consists  in  successively 
halving  the  region  or  one  part  of  the  region  which 
contains  an  infinity  of  points,*  we  arrive  at  what  is 
required, — namely,  the  conclusion  that  there  is  a 
point  such  that  there  is  another  point  in  any  neigh- 

*  This  method  was  first  used  by  Bernard  Bolzano  in  1817. 


14  INTRODUCTION 

bourhood  of  it,  that  is  to  say,  that  there  is  a  so- 
called  "point  of  condensation," — when,  and  only 
when,  we  have  proved  that  every  infinite  ''sum" 
such  that  the  sum  of  any  finite  number  of  its  terms 
does  not  exceed  some  given  finite  number  defines  a 
(rational  or  irrational)  number.  The  geometrical 
analogue  of  this  proposition  may  possibly  be  claimed 
to  be  evident ;  but  if  our  ideal  in  the  theory  of 
functions — which  had,  even  in  Weierstrass's  time, 
been  regarded  for  long  as  a  justified,  and  even  as  a 
partly  attained,  ideal — is  to  found  this  theory  on  the 
conception  of  number  alone,*  this  proposition  leads  to 
the  considerations  out  of  which  a  theory  of  irrational 
numbers  such  as  Weierstrass's  is  built.  The  theorem 
on  the  existence  of  at  least  one  point  of  condensa- 
tion was  proved  by  Weierstsass  by  the  method  of 
successive  subdivisions,  and  was  specially  emphasized 
by  him. 

Weierstrass,  in  the  introduction  to  his  lectures  on 

/       analytic  functions,  emphasized  that,  when  we  have 

admitted  the   notion   of  whole    number,    arithmetic 

needs  no  further  postulate,  but  can  be  built  up  in  a 

purely  logical  fashion,  and  also  that  the  notion  of  a 

"  The  separation  of  analysis  from  geometry,  which  appeared  in  the 
work  of  Lagrange,  Gauss,  Cauchy,  and  Bolzano,  was  a  consequence  of 
the  increasing  tendency  of  mathematicians  towards  logical  exactitude 
in  defining  their  conceptions  and  in  making  their  deductions,  and,  con- 
sequently, in  discovering  the  limits  of  validity  of  their  conceptions  and 
methods.  However,  the  true  connexion  between  the  founding  of 
analysis  on  a  purely  arithmetical  basis — "  arithmetization,"  as  it  has  been 
called — and  logical  rigour,  can  only  be  definitely  and  convincingly 
shown  after  the  comparatively  modern  thesis  is  proved  that  all  the  con- 
cepts (including  that  of  number)  of  pure  mathematics  are  wholly  logical. 
And  this  thesis  is  one  of  the  most  important  consequences  to  which  the 
theory  whose  growth  we  are  describing  has  forced  us. 


INTRODUCTION  15 

one-to-one  correspondence  is  fundamental  in  count- 
ing. But  it  is  in  his  purely  arithmetical  introduction 
of  irrational  numbers  that  his  great  divergence  from 
precedent  comes.  This  appears  from  a  consideration 
of  the  history  of  incomrtiensurables. 

The  ancient  Greeks  discovered  the  existence  of  in- 
commensurable  geometrical  magnitudes,  and  there- 
fore   grew   to    regard    arithmetic   and    geometry   as 
sciences    of   which    the    analogy  had    not    a  logical 
basis.      This  view  was  also  probably  due,  in  part  at 
least,   to   an  attentive  consideration   of  the   famous 
arguments    of   Zeno.     Analytical    geometry   practi- 
cally identified  geometry  with  arithmetic  (or  rather 
with    arithmetica    universalis)^    and,   before    Weier- 
strass,     the     introduction    of  irrational     'Miumber" 
was,    explicitly    or    implicitly,    geometrical.       The 
view  that  number  has  a  geometrical  basis  was  taken 
by  Newton  and  most  of  his  successors.      To  come 
to    the     nineteenth     century,      Cauchy     explicitly 
adopted  the  same  view.      At  the   beginning  of  his 
Cours  d' analyse  of   1821,    he  defined  a   "  Hmit "   as 
follows:    ''When    the    successive  values    attributed 
to   a   variable    approach   a    fixed    value   indefinitely 
so    as    to    end    by  differing   from    it   as   little  as   is 
wished,  this  fixed  value  is  called  the   '  limit '  of  all 
the  others  "  ;  and  remarked  that  "  thus  an  irrational 
number  is  the  Hmit  of  the  various  fractions  which 
furnish  more  and  more  approximate  values  of  it." 
If    we    consider — as,    however,    Cauchy    does     not 
appear  to  have  done,  although  many  others  have — 
the    lattei*    statement    as    a    definition,   so    that  an 


1 6  INTRODUCTION 

"irrational"  number  is  defined  to  be  the  limit  of 
certain  sums  of  rational  numbers,  we  presuppose 
that  these  sums  have  a  limit.  In  another  place 
Cauchy  remarked,  after  defining  a  series  u^,  u^, 
u^y  .  .  ,  to  be  convergent  if  the  sum  J„  =  2^0 +  ^1  +  ^2 
+  .  .  .  +2^«-i,  for  values  of  n  always  increasing, 
approaches  indefinitely  a  certain  limit  s,  that,  "by 
the  above  principles,  in  order  that  the  series 
^oj  ^^1'  ^'i^  '  '  '  ^^y  ^^  convergent,  it  is  necessary 
and  sufficient  that  increasing  values  of  n  make  the 
sum  s,,  converge  indefinitely  towards  a  fixed  limit 
s  ;  in  other  words,  it  is  necessary  and  sufficient  that, 
for  infinitely  great  values  of  n,  the  sums  s^,  j^^+i, 
s,,^2y  '  •  '  differ  from  the  limit  s,  and  consequently 
from  one  another,  by  infinitely  small  quantities." 
Hence  it  is  necessary  and  sufficient  that  the  different 
sums  u,,  +  u,,^i-{-  .  .  .  +2/,,+;;,,  for  different  m's,  end, 
when  n  increases,  by  obtaining  numerical  values  con- 
stantly differing  from  one  another  by  less  than  any 
assigned  number. 

If  we  know  that  the  sums  s,,  have  a  limit  s,  we 
can  at  once  prove  the  necessity  of  this  condition  ; 
but  its  sufficiency  (that  is  to  say,  if,  for  any  assigned 
positive  rational  6,  an  integer  n  can  always  be  found 
such  that 

where  r  is  any  integer,  then  a  limit  s  exists)  re- 
quires a  previous  definition  of  the  system  of  real 
numbers,  of  which  the  supposed  limit  is  to  be  one. 
For  it  is  evidently  a  vicious  circle  to  define  a  real 


INTRODUCTION  17 

number  as  the  limit  of  a  "convergent"  series,  as 
the  above  definition  of  what  we  mean  by  a  ''con- 
vergent "  series — a  series  which  Jias  a  Hmit — in- 
volves (unless  we  limit  ourselves  to  rational  limits) 
a  previous  definition  of  what  we  mean  by  a  "  real 
number."  * 

It  seems,  perhaps,  evident  to  "intuition"  that, 
if  we  lay  off  lengths  j^,  j,,+i,  .  .  .,  for  which  the 
above  condition  is  fulfilled,  on  a  straight  line,  that 
a  (commensurable  or  incommensurable)  "limiting" 
length  s  exists  ;  and,  on  these  grounds,  we  seem  to 
be  justified  in  designating  Cauchy's  theory  of  real 
number  as  geometrical.  But  such  a  geometrical 
theory  is  not  logically  convincing,  and  Weierstrass 
showed  that  it  is  unnecessary,  by  defining  real 
numbers  in  a  manner  which  did  not  depend  on  a 
process  of  "going  to  the  limit." 

To  repeat  the  point  briefly,  we  have  the  following 
logical  error  in  all  would-be  arithmetical  f  pre- 
Weierstrassian  introductions  of  irrational  numbers  : 
we  start  with  the  conception  of  the  system  of 
rational  numbers,  we  define  the  "sum"  (a  limit  of 
a  sequence  of  rational  numbers)  of  an  infijiite  series 
of  rational  numbers,  and  then  raise  ourselves  to  the 
conception  of  the  system  of  real  numbers  which  are 
got  by  such  means.  The  error  lies  in  overlooking 
the  fact  that  the  ' '  sum  "  {b)  of  the  infinite  series  of 

*  On  the  attempts  of  Bolzano,  Hankel,  and  Stolz  to  prove  arithmetic- 
ally, without  an  arithmetical  theory  of  real  numbers,  the  sufficiency  of 
the  above  criterion,  see  Ostwald^s  A'iassiker,  No.  153,  pp.  42,  95,  107. 

t  It  must  be  remembered  that  Cauchy's  theory  was  not  one  of  these, 
Cauchy  did  not  attempt  to  define  real  numbers  arithmetically,  but 
simply  presupposed  their  existence  on  geometrical  grounds. 

2 


V 


> 


ig  INTRODUCTION 

rational  numbers  can  only  be  defined  when  we  have 
already  defined  the  real  numbers,  of  which  b  is  one. 
"  I  believe,"  said  Cantor,*  a propos  of  Weierstrass's 
theory,  "that  this  logical  error,  which  was  first 
avoided  by  Weierstrass,  escaped  notice  almost 
universally  in  earlier  times,  and  was  not  noticed  on 
the  ground  that  it  is  one  of  the  rare  cases  in  which 
actual  errors  can  lead  to  none  of  the  more  important 
mistakes  in  calculation." 

Thus,  we  must  bear  in  mind  that  an  arithmetical 
theory  of  irrationals  has  to  define  irrational  numbers 
not  as  "limits"  (whose  existence  is  not  always 
beyond  question)  of  certain  infinite  processes,  but 
in  a  manner  prior  to  any  possible  discussion  of  the 
question  in  what  cases  these  processes  define  limits 
at  all. 

With  Weierstrass,  a  number  was  said  to  be 
"determined"  if  we  know  of  what  elements  it  is 
composed  and  how  many  times  each  element 
occurs  in  it.  Considering  numbers  formed  with 
the  principal  unit  and  an  infinity  of  its  aHquot  parts, 
Weierstrass  called  any  aggregate  whose  elements 
and  the  number  (finite)  of  times  each  element 
occurs  in  it  f  are  known  a  (determined)  "  numerical 
quantity  "  {Zahlengrosse).  An  aggregate  consisting 
of  a  finite  number  of  elements  was  regarded  as  equal 
to  the  sum  of  its  elements,  and  two  aggregates  of  a 
finite  number  of  elements  were  regarded  as  equal 
when  the  respective  sums  of  their  elements  are  equal. 

*  Math.  Ann.,  vol.  xxi,  1883,  p.  566. 

t  It  is  not  implied  that  the  given  elements  are  finite  in  number. 


INTRODUCTION  19 

A  rational  number  r  was  said  to  be  contained  in 
a  numerical  quantity  a  when  we  can  separate  from 
a  a  partial  aggregate  equal  to  r.  A  numerical 
quantity  a  was  said  to  be  ''finite"  if  we  could 
assign  a  rational  number  R  such  that  every  rational 
number  contained  in  a  is  smaller  than  R.  Two 
numerical  quantities  <:^,  b  were  said  to  be  "equal," 
when  every  rational  number  contained  in  a  is  con- 
tained in  b^  and  vice  versa.  When  a  and  b  are  not 
equal,  there  is  at  least  one  rational  number  which 
is  either  contained  in  a  without  being  contained  in 
^,  or  vice  versa  :  in  the  first  case,  a  was  said  to  be 
' '  greater  than  "  b  ;  in  the  second,  a  was  said  to  be 
' '  less  than  "  b. 

VVeierstrass  called  the  numerical  quantity  c  de- 
fined by  {i.e.  identical  with)  the  aggregate  whose 
elements  are  those  which  appear  in  a  or  ^,  each  of 
these  elements  being  taken  a  number  of  times  equal 
to  the  number  of  times  in  which  it  occurs  in  a 
increased  by  the  number  of  times  in  which  it  occurs 
in  b^  the  ''  stmi''  of  a  and  b.  The  ''product'"  of 
a  and  b  was  defined  to  be  the  numerical  quantity 
defined  by  the  aggregate  whose  elements  are  ob- 
tained by  forming  in  all  possible  manners  the  product 
of  each  element  of  a  and  each  element  of  b.  In  the 
same  way  was  defined  the  product  of  any  finite 
number  of  numerical  quantities. 

The  "sum"  of  an  infinite  number  of  numerical 
quantities  a,  ^,  .  .  .  was  then  defined  to  be  the 
aggregate  {s)  whose  elements  occur  in  one  (at  least) 
of  ^,  ^,  .  .  .,  each  of  these  elements  e  being  taken 


20  INTRODUCTION 

a  number  of  times  (;?)  equal  to  the  number  of  times 
that  it  occurs  in  a,  increased  by  the  number  of  times 
that  it  occurs  in  b^  and  so  on.  In  order  that  s  be 
finite  and  determined,  it  is  necessary  that  each  of  the 
elements  which  occurs  in  it  occurs  a  finite  number  of 
times,  and  it  is  necessary  and  sufficient  that  we  can 
assign  a  number  N  such  that  the  sum  of  any  finite 
number  of  the  quantities  a,  b^  .  .  .  \s  less  than  N. 
,  Such  is  the  principal  point  of  Weierstrass's  theory 
y/of  real  numbers.  It  should  be  noticed  that,  with 
Weierstrass,  the  new  numbers  were  aggregates  of 
the  numbers  previously  defined  ;  and  that  this  view, 
which  appears  from  time  to  time  in  the  better  text- 
books, has  the  important  advantage  which  was  first 
sufficiently  emphasized  by  Russell.  This  advantage 
is  that  the  existence  of  limits  can  be  proved  in 
such  a  theory.  That  is  to  say,  it  can  be  proved  by 
actual  construction  that  there  is  a  number  which  is 
the  limit  of  a  certain  series  fulfilling  the  condition 
of  ' '  finiteness  "or  "  convergency. "  When  real 
numbers  are  introduced  either  without  proper  defini- 
tions, or  as  "creations  of  our  minds,"  or,  what  is 
far  worse,  as  "signs,"*  this  existence  cannot  be 
proved. 

If  we  consider  an  infinite  aggregate  of  real 
numbers,  or  comparing  these  numbers  for  the  sake 
of  picturesqueness  with  the  points  of  a  straight 
Hne,  an  infinite  "point-aggregate,"  we  have  the 
theorem  :  There  is,  in  this  domain,  at  least  one  point 
such  that  there  is  an  infinity  of  points  of  the  aggre- 

*   C/;  Juurdain,  Math.  Gazette,  ]iin.  1908,  vol.  iv,  pp.  201-209. 


INTRODUCTION  21 

gate  in  any,  arbitrarily  small,  neighbourhood  of  it. 
Weierstrass's  proof  was,  as  we  have  mentioned, 
by  the  process,  named  after  Bolzano  and  him, 
of  successively  halving  any  one  of  the  intervals 
which  contains  an  infinity  of  points.  This  process 
defines  a  certain  numerical  magnitude,  the  ''point 
of  condensation  "  {Hdufungss telle)  in  question.  An 
analogfous  theorem  holds  for  the  two-dimensional 
region  of  complex  numbers. 

Of  real  numerical  magnitudes  x^  all  of  which  are 
less  than  some  finite  number,  there  is  an  ''upper 
limit,"  which  is  defined  as  :  A  numerical  magnitude 
G  which  is  not  surpassed  in  magnitude  by  any  x  and 
is  such  that  either  certain  x's  are  equal  to  G  or 
certain  x's  lie  within  the  arbitrarily  small  interval 
(G,  .  .  . ,  G  —  (5),  the  end  G  being  excluded.  Ana- 
logously for  the  "  lower  limit  "^. 

It  must  be  noticed  that,  if  we  have  ^  finite 
aggregate  of  x's,  one  of  these  is  the  upper  limit, 
and,  if  the  aggregate  is  infinite,  one  of  them  may 
be  the  upper  hmit.  In  this  case  it  need  not  also, 
but  of  course  may,  be  a  point  of  condensation.  If 
none  of  them  is  the  upper  limit,  this  limit  (whose 
existence  is  proved  similarly  to  the  existence  of  a 
point  of  condensation,  but  is,  in  addition,  unique] 
is  a  point  of  condensation.  Thus,  in  the  above 
explanation  of  the  term  "upper  limit,"  we  can 
replace  the  words  "either  certain  ,r's "  to  "being 
excluded"  by  "  certain  ;r's  lie  in  the  arbitrary  small 
interval  (G,  .  .  .,  G  —  S),  the  end  G  being  i^icluded.'' 

The    theory  of   the    upper   and  lower  limit  of  a 


22  INTRODUCTION 

(general  or  '' Dirichlet's ")  real  one-valued  function 
of  a  real  variable  was  also  developed  and  emphasized 
by  Weierstrass,  and  especially  the  theorem  :  If  G  is 
the  upper  limit  of  those  values  o{  y=f{x)'^  which 
belong  to  the  values  of  x  lying  inside  the  interval 
from  a  to  h,  there  is,  in  this  interval,  at  least  one 
point  ;i  =  X  such  that  the  upper  hmit  of  the  j's 
which  belong  to  the  ;r's  in  an  arbitrarily  small 
neighbourhood  of  X  is  G  ;  and  analogously  for  the 
lower  limit. 

If  the  j^-value  corresponding  to  ;r=X  is  G,  the 
upper  limit  is  called  the  "maximum"  of  the  j^'s 
and,  if  j?^=/"(-^)  is  a  continuous  function  of  x^  the 
upper  limit  is  a  maximum  ;  in  other  words,  a  con- 
tinuo2is  function  attains  its  upper  and  lower  limits. 
That  a  continuous  function  also  takes  at  least  once 
every  value  between  these  limits  was  proved  by 
Bolzano  (1817)  and  Cauchy  (1821),  but  the  Weier- 
strassian  theory  of  real  numbers  first  made  these 
proofs  rigorous,  f 

It  is  of  the  utmost  importance  to  realize  that, 
whereas  until  Weierstrass's  time  such  subjects  as 
the  theory  of  points  of  condensation  of  an  infinite 
aggregate  and  the  theory  of  irrational  numbers, 
on  which  the   founding  of  the   theory   of  functions 

*  Even  \i  y  is  finite  for  every  single  x  of  the  interval  a^x^b,  all 
these  jj/'s  need  not  be,  in  absolute  amount,  less  than  some  finite  number 
(for  example,  f{x)=\\x  for  jf>0, /(o)=o,  in  the  interval  o^  ^^i), 
but  if  they  are  (as  in  the  case  of  the  sum  of  a  uniformly  convergent 
series),  these  j/'s  have  a  finite  upper  and  lower  limit  in  the  sense  defined. 

t  There  is  another  conception  (due  to  Cauchy  and  P.  du  Bois- 
Reymond),  allied  to  that  of  upper  and  lower  limit.  _  With  every  infinite 
aggregate,  there  are  (attained)  upper  and  lower  points  of  condensation, 
which  we  may  call  by  the  Latin  name  "  Limites" 


INTRODUCTION  23 

depends,  were  hardly  ever  investigated,  and  never 
with  such  important  results,  Weierstrass  carried 
research  into  the  principles  of  arithmetic  farther 
than  it  had  been  carried  before.  But  we  must  also 
realize  that  there  were  questions,  such  as  the  nature  < 
of  whole  number  itself,  to  which  he  made  no  valuable 
contributions.  These  questions,  though  logically 
the  first  in  arithmetic,  were,  of  course,  historically 
the  last  to  be  dealt  with.  Before  this  could  happen, 
arithmetic  had  to  receive  a  development,  by  means  ^y 
of  Cantor's  discovery  of  transfinite  numbers,  into  a  ^ 
theory  of  cardinal  and  ordinal  numbers,  both  finite 
and  transfinite,  and  logic  had  to  be  sharpened,  as 
it  was  by  Dedekind,  Frege,  Peano  and  Russell — to 
a  great  extent  owing  to  the  needs  which  this  theory 
made  evident.  y 

V 

Georg  Ferdinand  Ludwig  Philipp  Cantor  was 
born  at  St  Petersburg  on  3rd  March  1845,  ^"d 
lived  there  until  1856;  from  1856  to  1863  he  lived 
in  South  Germany  (Wiesbaden,  Frankfurt  a.  M., 
and  Darmstadt);  and,  from  autumn  1863  to  Easter 
1869,  in  Berlin.  He  became  Privatdocent  at  Halle 
a.  S.  in  1869,  extraordinary  Professor  in  1872,  and 
ordinary  Professor  in  1879.*  When  a  student  at 
Berlin,  Cantor  came  under  the  influence  of  Weier- 
strass's    teaching,    and    one   of   his   first    papers   on 

*  Those  memoirs  of  Cantor's  that  will  be  considered  here  more 
particularly,  and  which  constitute  by  far  the  greater  part  of  his  writings, 
are  contained  in  :  Journ.  fiir  Alath.^  vols.  Ixxvii  and  Ixxxiv,  1874  and 
1878;  Math.  Ann.,  vol.  iv,  1871,  vol.  v,  1872,  vol.  xv,  1879,  vol.  xvii, 
j88o,  vol.  XX,  1882,  vol,  xxi,  1883, 


24  INTRODUCTION 

mathematics  was  partly  occupied  with  a  theory  of 
irrational  numbers,  in  which  a  sequence  of  numbers 
satisfying  Cauchy's  condition  of  convergence  was 
I  used  instead  of  Weierstrass's  complex  of  an  infinity 
of  elements  satisfying  a  condition  which,  though 
equivalent  to  the  above  condition,  is  less  convenient 
for  purposes  of  calculation. 

This  theory  was  exposed  in  the  course  of  Cantor's 
researches  on  trigonometrical  series.  One  of  the 
problems  of  the  modern  theory  of  trigonometrical 
series  was  to  establish  the  uniqueness  of  a  trigono- 
metrical development.  Cantor's  investigations  re- 
lated to  the  proof  of  this  uniqueness  for  the  most 
general  trigonometrical  series,  that  is  to  say,  those 
trigonometrical  series  whose  coefficients  are  not 
necessarily  supposed  to  have  the  (Fourier's)  integral 
form. 

In  a  paper  of  1870,  Cantor  proved  the  theorem 
that,  if 

a^,  a^,  .  .  .,  a^,  .  .  .  and  b^,  b.,  .  .  .,  b^,  .  .  . 

are  two  infinite  series  such  that  the  limit  of 

a^  sin  vx-{-b^,  cos  vx, 


for  every  value  of  x  which  lies  in  a  given  interval 
{a<x<b)  of  the  domain  of  real  magnitudes,  is  zero 
with  increasing  v,  both  a^  and  b^  converge,  with 
increasing  i/,  to  zero.  This  theorem  leads  to  a 
criterion  for  the  convergence  of  a  trigonometrical 
series 

\b^^a^^\Xix^b^QO'6x-\- . . .  -\- a,,'=^\\\vx -\- b ^QO^vx -\-  .  .  ., 


INTRODUCTION  25 

that  Riemann  proved  under  the  supposition  of  the 
integral  form  for  the  coefficients.  In  a  paper  im- 
mediately following  this  one,  Cantor  used  this 
theorem  to  prove  that  there  is  only  one  representation 
oi  f{x)  in  the  form  of  a  trigonometrical  series  con- 
vergent for  every  value  of  x,  except,  possibly,  a 
finite  number  of  x's  ;  if  the  sums  of  two  trigono- 
metrical series  differ  for  a  finite  number  of  ;tr's,  the 
forms  of  the  series  coincide. 

In  1 87 1,  Cantor  gave  a  simpler  proof  of  the 
uniqueness  of  the  representation,  and  extended  this 
theorem  to  :  If  we  have,  for  every  value  of  x,  a 
convergent  representation  of  the  value  o  by  a 
trigonometrical  series,  the  coefficients  of  this  re- 
presentation are  zero.  In  the  same  year,  he  also 
gave  a  simpler  proof  of  his  first  theorem  that,  if 
lim  {a^  sin  vx-\-b^  cos  vx)  =  o  for  a<x<b,  then  both 
lim  a^,  and  lim  b^  are  zero. 

In  November  1871,  Cantor  further  extended  his 
theorem  by  proving  that  the  convergence  or  equality 
of  the  sums  of  trigonometrical  series  may  be  re- 
nounced for  certain  infinite  aggregates  of  x's  in  the  / 
interval  o  .  .  .  27r  without  the  theorem  ceasing  to 
hold.  To  describe  the  structure  that  such  an 
aggregate  may  have  in  this  case,  Cantor  began 
with  "some  explanations,  or  rather  some  simple 
indications,  intended  to  put  in  a  full  light  the 
different  manners  in  which  numerical  magnitudes, 
in  number  finite  or  infinite,  can  behave,"  in  order 
to  make  the  exposition  of  the  theorem  in  question 
as  short  as  possible. 


26  INTRODUCTION 

The  system  A  of  rational  numbers  (including  o) 
serves  as  basis  for  arriving  at  a  more  extended 
notion  of  numerical  magnitude.  The  first  general- 
ization with  which  we  meet  is  when  we  have  an 
infinite  sequence 

(i)  «i,  a^,  .  .  .,  a,,  .  .  , 

of  rational  numbers,  given  by  some  law,  and  such 
that,  if  we  take  the  positive  rational  number  e  as 
small  as  we  wish,  there  is  an  integer  n^  such  that 

(2)  I  a,,+,„-a,,  I  <e     (^>^i), 

whatever  the  positive  integer  m  is.*  This  property 
Cantor  expressed  by  the  words,  ''the  series  (i) 
has  a  determined  limit  ^,"  and  remarked  particularly 
that  these  words,  at  that  point,  only  served  to 
enunciate  the  above  property  of  the  series,  and, 
just  as  we  connect  (i)  with  a  special  sign  d,  we 
must  also  attach  different  signs  l?\  b'\  .  .  .,  to 
different  series  of  the  same  species.  However, 
because  of  the  fact  that  the  "limit"  may  be 
supposed  to  be  previously  defined  as  :  the  number 
(if  such  there  be)  d  such  that  \b  —  a,\  becomes  in- 
finitely small  as  v  increases,  it  appears  better  to 
avoid  the  word  and  say,  with  Heine,  in  his  ex- 
position of  Cantor's  theory,  the  series  (a^)  is  a 
''  number-series,"  or,  as  Cantor  afterwards  expressed 
it,  (a^)  is  a  "fundamental  series." 

*  It  may  be  proved  that  this  condition  (2)  is  necessary  and  sufficient 
that  the  sum  to  infinity  of  the  series  corresponding  to  the  sequence  (l) 
should  be  a  "  finite  numerical  mai^nitude  "  in  Weierstrass's  sense  ;  and 
consequently  Cantor's  theory  of  irrational  numbers  has  been  described 
as  a  happy  modification  of  Weierstrass's, 


INTRODUCTION  27 

Let  a  second  series 

(i')  a\,  a\,  .  .  .,  a\,,  .  .  . 

have  a  determined  limit  d\  we  find  that  (i)  and 
(i')  have  always  one  of  the  three  relations,  which 
exclude  one  another:  (a)  a„  —  a'„  becomes  infinitely 
small  as  7i  increases  ;  (d)  from  a  certain  n  on,  it 
remains  always  greater  than  e,  where  e  is  positive 
and  rational  ;  (c)  from  a  certain  n  on,  it  remains 
always  less  than  —e.  In  these  cases  we  say, 
respectively, 

/;  =  ^',      /;>//,      or      /?<d'. 

Similarly,  we  find  that  (i)  has  only  one  of  the 
three  relations  with  a  rational  number  a  :  (a)  a,^  —  a 
becomes  infinitely  small  as  n  increases  ;  (d)  from 
a  certain  n  on,  it  remains  always  greater  than  e  ; 
(c)  from  a  certain  n  on,  it  remains  less  than  —  e. 
We  express  this  by 

l?  =  a,      d>a,      or     l?<a, 

respectively.  Then  we  can  prove  that  /;  —  a„  becomes 
infinitely  small  as  n  increases,  which,  consequently, 
justifies  the  name  given  to  I?  of  ''limit  of  the 
series  (i). " 

Denoting  the  totality  of  the  numerical  magnitudes 
d  by  B,  we  can  extend  the  elementary  operations 
with  the  rational  numbers  to  the  systems  A  and  B 
united.      Thus  the  formulae 

l?±d'  =  y\     bb'  =  b'\     b\b'  =  b" 


28  INTRODUCTION 

express  that  the  relations 

lim  {a„^a  ,,  -  a" ,^  =  o,      \\m  (a,,a  „  -  a" ,)  =  o, 

hold    respectively.        We    have    similar    definitions 
when  one  or  two  of  the  numbers  belong  to  A. 

The  system  A  has  given  rise  to  B  ;  by  the  same 
process  B  and  A  united  give  rise  to  a  third  system 
C.      Let  the  series 

(3)  K    ^2.    •    •    •'    ^v,    .    .    . 

be  composed  of  numbers  from  A  and  B  (not  all 
from  A),  and  such  that  |  b,,^„,  —  bn  \  becomes  in- 
finitely small  as  n  increases,  whatever  in  is  (this 
condition  is  determined  by  the  preceding  definitions), 
then  (3)  is  said  to  have  '*a  determined  limit  ^." 
The  definitions  of  equality,  inequality,  and  the 
elementary  operations  with  the  members  of  C,  or 
with  them  and  those  of  B  and  A,  are  analogous  to 
the  above  definitions.  Now,  whilst  B  and  A  are  such 
that  we  can  equate  each  <^  to  a  <^,  but  not  inversely, 
we  can  equate  each  ^  to  a  <;,  and  inversely.  ' '  Although 
thus  B  and  C  can,  in  a  certain  measure,  be  regarded 
as  identical,  it  is  essential  in  the  theory  here  ex- 
posed, according  to  which  the  numerical  magnitude, 
not  having  in  general  any  objectivity  at  first,*  only 
appears  as  element  of  theorems  which  have  a  certain 
objectivity  (for  example,  of  the  theory  that  the 
numerical  magnitude  serves  as  limit  for  the  corre- 
sponding series),  to  maintain  the  abstract  distinction 

*  This  is  connected  with  Cantor's  formalistic  view  of  real  numbers 
(see  below). 


INTRODUCTION  29 

between  B  and  C,  and  also  that  the  equivalence  of 
b  and  b'  does  not  mean  their  identity,  but  only 
expresses  a  determined  relation  between  the  series 
to  which  they  refer." 

After  considering  further  systems  C,  D,  .  .  .,  L 
of  numerical  magnitudes  which  arise  successively, 
as  B  did  from  A  and  C  from  A  and  B,  Cantor  dealt 
with  the  relations  of  the  numerical  magnitudes  with 
the  metrical  geometry  of  the  straight  line.  If  the 
distance  from  a  fixed  point  O  on  a  straight  line  has 
a  rational  ratio  with  the  unit  of  measure,  it  is 
expressed  by  a  numerical  magnitude  of  the  system 
A  ;  otherwise,  if  the  point  is  known  by  a  con- 
struction, we  can  always  imagine  a  series  such  as 
(i)  and  having  with  the  distance  in  question  a 
relation  such  that  the  points  of  the  straight  line  to 
which  the  distances  a^^  a^^  ■  ■  -,  ^v,  •  •  •  refer 
approach,  ad  infinitum^  as  v  increases,  the  point  to 
be  determined.  We  express  this  by  saying  :  The 
distance  from  the  point  to  be  determined  to  the 
point  O  is  equal  to  b^  where  b  is  the  numerical 
magnitude  corresponding  to  the  series  (i).  VVe  can 
then  prove  that  the  conditions  of  equivalence, 
majority,  and  minority  of  known  distances  agree 
with  those  of  the  numerical  magnitudes  which 
represent  these  distances. 

It  now  follows  without  difficulty  that  the  numerical 
magnitudes  of  the  systems  C,  D,  .  .  .,  are  also 
capable  of  determining  the  known  distances.  But, 
to  complete  the  connexion  we  observe  between  the 
systems  of  numerical  magnitudes  and  the  geometry 


30  INTRODUCTION 

of  the  straight  line,  an  axiom  must  still  be  added, 
which  runs  :  To  each  numerical  magnitude  belongs 

falso,  reciprocally,  a  determined  point  of  the  straight 
line  whose  co-ordinate  is  equal  to  this  numerical 
magnitude.*  This  theorem  is  called  an  axiom,  for 
in  its  nature  it  cannot  be  demonstrated  generally. 
It  also  serves  to  give  to  the  numerical  magnitudes 
a  certain  objectivity,  of  which,  however,  they  are 
completely  independent. 

We  consider,  now,  the  relations  which  present  them- 
selves when  we  are  given  a  finite  or  infinite  system  of 
numerical  magnitudes,  or  "points,"  as  we  may  call 
them  by  what  precedes,  with  greater  convenience. 

If  we  are  given  a  system  (P)  of  points  in  a  finite 
interval,  and  understand  by  the  word  "  limit-point  " 
{Grenzpunkt)  a  point  of  the  straight  line  (not 
necessarily  of  P)  such  that  in  any  interval  within 
which  this  point  is  contained  there  is  an  infinity  of 
points  of  P,  we  can  prove  VVeierstrass's  theorem 
that,  if  P  is  infinite,  it  has  at  least  one  limit-point. 
Every  point  of  P  which  is  not  a  limit-point  of  P 
was  called  by  Cantor  an  "  isolated  "  point. 

Every  point,  then,  of  the  straight  line  either  is  or 
is  not  a  limit-point  of  P  ;  and  we  have  thus  defined, 
at  the  same  time  as  P,  the  system  of  its  limit-points, 
which  may  be  called  the  "first  derived  system" 
{erste  Ableitung)  V .  If  P'  is  not  composed  of  a 
finite  number  of  points,  we  can  deduce,  by  the  same 


*  To  each  numerical  magnitude  belongs  a  determined  point,  but  to 
each  point  are  related  as  co-ordinates  numberless  equal  numerical 
magnitudes. 


INTRODUCTION  3I 

process,  a  second  derived  system  P''  from  F  ;  and, 
by  V  analogous  operations,  we  arrive  at  the  notion 
of  a  i/th  system  P^"^  derived  from  P,  If,  for 
example,  P  is  composed  of  all  the  points  of  a  line 
whose  abscissae  are  rational  and  comprised  between 
O  and  I  (including  these  limits  or  not),  P'  is  com- 
posed of  all  the  points  of  the  interval  (o  .  .  .  i), 
including  these  limits  ;  and  P',  P",  ...  do  not 
differ  from  P.  If  P  is  composed  of  the  points  whose 
abscissae  are  respectively 

I,    1/2,   1/3,    .    ,    .,    \\v  .    .    ., 

P'  is  composed  of  the  single  point  o,  and  derivation 
does  not  give  rise  to  any  other  point.  It  may 
happen — and  this  case  alone  interests  us  here — 
— that,  after  v  operations,  P^")  is  composed  of  a  finite 
number  of  points,  and  consequently  derivation  does 
not  give  rise  to  any  other  system.  In  this  case 
the  primitive  P  is  said  to  be  of  the  ^^  vX\\  species 
{Art);'  and  thus  Y\  P",  ...  are  of  the  (t/-i)th, 
(i/— 2)th,  .  .  .  species  respectively. 

The  extended    trigonometrical    theorem  is   now  : 
If  the  equation 

0  =  J^Q  +  <a:j  sin. r -1-^1  cos  ^'+  .  .  .    -\-a^^\\\vX 

+  ^^cos  vx-\-  .  .  , 

is  satisfied  for  all  values  of  x  except  those  which 
correspond  to  the  points  of  a  system  P  of  the  i/th 
species,  where  v  is  an  integer  as  great  as  is  pleased, 
in  the  interval  (o  .  .  .  27r),  then 

b^  =  o,      c\  =  b,  =  o. 


32  INTRODUCTION 

Further  information  as  to  the  conthiuation  of 
these  researches  into  derivatives  of  point-aggregates 
was  given  in  the  series  of  papers  which  Cantor 
began  in  1879  under  the  title  "  Ueber  unendHche, 
lineare  Punktmannichfaltigkeiten."  Although  these 
papers  were  written  subsequently  to  Cantor's  dis- 
covery (1873)  of  the  conceptions  of  "  enumerability  " 
{Abzdhlbarkeit)  and  "power"  {Miichtigkeit),  and 
these  conceptions  formed  the  basis  of  a  classification 
of  aggregates  which,  together  with  the  classification 
by  properties  of  the  derivatives  to  be  described 
directly,  was  dealt  with  in  these  papers,  yet,  since, 
by  Cantor's  own  indications,*  the  discovery  even  of 
derivatives  of  definitely  infinite  order  was  made  in 
1 87 1,  we  shall  now  extract  from  these  papers -the 
parts  concerning  derivatives. 

A  point-aggregate  P  is  said  to  be  of  the   ''first 
kind"  (Gattung)  and  i^th   ''species"  if  P^''^  consists 
of  merely  a  finite  aggregate  of  points  ;  it  is  said  to 
be  of  the  ' '  second  kind  "  if  the  series 
p'     p"  p(..) 

is  infinite.  All  the  points  of  P",  Y" ,  ...  are  always 
points  of  P',  while  a  point  of  P'  is  not  necessarily  a 
point  of  P. 

*  In  1880,  Cantor  wrote  of  the  "  dialectic  generation  of  conceptions, 
which  always  leads  farther  and  yet  remains  free  from  all  arbitrariness, 
necessary  and  logical,"  of  the  transfinite  series  of  indices  of  derivatives. 
"I  arrived  at  this  ten  years  ago  [this  was  written  in  May  1880]  ;  on 
the  occasion  of  my  exposition  of  the  number-conception,  I  did  not 
refer  to  it."  And  in  a  letter  to  me  of  31st  August  1905,  Professor 
Cantor  wrote:  "Was  die  transfiniten  Ordnungszahlen  betriftt,  ist  es 
mir  wahrscheinlich,  dass  ich  schon  1871  eine  Voistellung  vcn  ihnen 
gehal:)t  habe.  Den  Begrifif  der  Abzahlbarkeit  bildete  ich  mir  erst 
1873." 


INTRODUCTION  33 

Some  or  all  of  the  points  of  a  continuous  *  interval 
(a  .  .  .  ^),  the  extreme  points  being  considered  as 
belonging  to  the  interval,  may  be  points  of  P  ;  if 
none  are,  ?  is  said  to  be  quite  outside  (a  .  .  .  /3).  If 
P  is  (wholly  or  in  part)  contained  in  (a  .  .  .  0),  a 
remarkable  case  may  present  itself  :  every  interval 
(y  .  .  .  ^)  in  it,  however  small,  may  contain  points 
of  P.  Then  P  is  said  to  be  '  *  everywhere  dense  " 
in  the  interval  (a  .  .  .  fi).  For  example,  (i)  the 
point-aggregate  whose  elements  are  all  the  points 
of  (a  .  .  .  /3),  (2)  that  of  all  the  points  whose 
abscissa?  are  rational,  and  (3)  that  of  all  the 
points  whose  abscissae  are  rational  numbers  of  the 
form  zL(2n-\-  i)/2"',  where  m  and  n  are  integers,  are 
everywhere  dense  in  (a  .  .  .  ^)-  It  results  from  this 
that,  if  a  point-aggregate  is  not  everywhere  dense 
in  (a  .  .  .  ^),  there  must  exist  an  interval  (y  .  .  .  ^) 
comprised  in  (a  .  .  .  /3)  and  in  which  there  is  no 
point  of  P.  Further,  if  P  is  everywhere  dense 
in  (a  .  .  .  /3),  not  only  is  the  same  true  for  P', 
but  P'  consists  of  all  the  points  of  («  .  .  .  jS).  We 
might  take  this  property  of  P'  as  the  definition 
of  the  expression  :  * '  P  is  everywhere  dense  in 
{a...,8y 

Such  a  P  is  necessarily  of  the  second  kind,  and 
hence  a  point-aggregate  of  the  first  kind  is  every- 
where dense  in  no  interval.  As  to  the  question 
whether    inversely    every    point-aggregate    of    the 

*  At  the  beginning  of  the  first  paper,  Cantor  stated:  "As  we  shall 
show  later,  it  is  on  this  notion  [of  derived  aggregate]  that  the  simplest 
and  completest  explanation  respecting  the  determination  of  a  continuum 
rests  "  (see  below). 

3 


34  INTRODUCTION 

second  kind  is  everywhere  dense  in  some  intervals, 
Cantor  postponed  it.  y 

Point-aggregates  of  the  first  kind  can,  as  we  have 
seen,  be  completely  characterized  by  the  notion  of 
derived  aggregate,  but  for  those  of  the  second  kind 
this  notion  does  not  suffice,  and  it  is  necessary  to 
give    it    an    extension   which    presents    itself   as    it 
were  of  its  own  accord  when  we  go  deeper  into  the 
question.      It  may  here  be  remarked  that  Paul  du 
Bois-Reymond  was  led  by  the  study  of  the  general 
theory  of  functions  to  a  partly  similar  development 
of  the  theory  of  aggregates,  and  an  appreciation  of 
its  importance  in  the  theory  of  functions.      In  1874, 
he  classified  functions    into  divisions,   according  to 
the  variations  of  the  functions  required  in  the  theory 
of  series   and   integrals  which   serve   for  the   repre- 
sentation   of     "arbitrary"     functions.        He    then 
considered    certain     distributions     of    singularities. 
An  infinite  aggregate  of  points  which  does  not  form 
a  continuous  line  may  be  either   such  that  in  any 
line,  however  small,  such  points  occur  (like  the  points 
corresponding  to  the  rational  numbers),  or  in  any 
part,  a  finite  line  in  which  are  none  of  those  points 
exists.      In  the  latter  case,  the  points  are  infinitely 
dense  on  nearing  certain  points  ;   "for  if  they  are 
infinite   in   number,    all    their    distances   cannot    be 
finite.        But    also    not    all    their    distances     in    an 
arbitrarily   small    line   can   vanish  ;    for,    if  so,    the 
first  case  would  occur.      So   their  distances  can  be 
zero   only  in   points,    or,   speaking   more   correctly, 
in    infinitely   small   lines."       Here    we    distinguish: 


INTRODUCTION  35 

(i)  The  points  k^  condense  on  nearing  a  finite 
number  of  points  k^  ;  (2)  the  points  k^  condense  at 
a  finite  number  of  points  k^,  .  .  .  Thus,  the 
roots  of  o  =  sin  i/jc  condense  near  x=o,  those  of 
o  =  sin  I /sin  i/x  near  the  preceding  roots,  .  .  . 
The  functions  with  such  singularities  fill  the  space 
between  the  "  common  "  functions  and  the  functions 
with  singularities  from  point  to  point.  Finally, 
du  Bois-Reymond  discussed  integration  over  such 
a  line.  In  a  note  of  1879,  he  remarked  that 
Dirichlet's  criterion  for  the  integrability  of  a  function 
is  not  sufficient,  for  we  can  also  distribute  intervals 
in  an  everywhere  dense  fashion  {pantacJiiscJi)  ;  that 
is  to  say,  we  can  so  distribute  intervals  D  on  the 
interval  ( —  tt  .  .  .  +  tt)  that  in  any  connected 
portion,  however  small,  of  (  — tt  .  .  .  +7r)  connected 
intervals  D  occur.  Let,  now,  ^(,r)  be  o  in  these 
D's  and  i  in  the  points  of  (  — tt  .  .  .  +7r)  not  covered 
by  D's  ;  then  (^{x)  is  not  integrable,  although  any 
interval  inside  ( —  tt  .  .  .  +  tt)  contains  lines  in 
which  it  is  continuous  (namely,  zero).  *'To  this 
distribution  of  intervals  we  are  led  when  we  seek 
the  points  of  condensation  of  infinite  order  whose  ex- 
istence i  announced  to  Professor  Cantor  years  ago." 
Consider  a  series  of  successive  intervals  on  the 
line  like  those  bounded  by  the  points  i,  1/2,  1/3, 
.  .  .,  i/r,  .  .  .  ;  in  the  interval  (1/1/ .  .  .  i/(i/+ i)) 
take  a  point-aggregate  of  the  first  kind  and  j/th 
species.  Now,  since  each  term  of  the  series  of 
derivatives  of  P  is  contained  in  the  preceding  ones, 
and  consequently  each  P^")  arises  from  the  preceding 


36  INTRODUCTION 

pC'^-i)  by  the  falling  away  (at  most)  of  points, — that 
is  to  say,  no  new  points  arise, — then,  if  P  is  of  the 
second  kind,  P'  will  be  composed  of  two  point- 
aggregates,  Q  and  R  ;  Q  consisting  of  those  points 
of  P  which  disappear  by  sufficient  progression  in 
the  sequence  P',  P'',  .  .  .,  P^"),  .  .  .,  and  R  of  the 
points  kept  in  all  the  terms  of  this  sequence.  In 
the  above  example,  R  consists  of  the  single  point 
zero.  Cantor  denoted  R  by  P(°°\  and  called  it 
"the  derived  aggregate  of  P  of  order  oo  (infinity)." 
The  first  derivative  of  P^°°)  was  denoted  by  p(~+i), 
and  so  on  for 


p(cxD+2)      p(co+3)      ^     ^     ^        p(o 


+v) 


Again,  P(~)  may  have  a  derivative  of  infinite  order 
which  Cantor  denoted  by  P^^°°^ ;  and,  continuing 
these  conceptual  constructions,  he  arrived  at  de- 
rivatives which  are  quite  logically  denoted  by 
p(woo+«)^  where  in  and  n  are  positive  integers.  But 
he  went  still  farther,  formed  the  aggregate  of 
common  points  of  all  these  derivatives,  and  got  a 
derivative  which  he  denoted  p(°°'^),  and  so  on  without 
end.      Thus  he  got  derivatives  of  indices 


S3 


1^0^   +1^100       i-   .   .   .    -tv^^y  ...    00  ,  ...   CO     J  .  .  . 

"Here  we  see  a  dialectic  generation  of  concep- 
tions,* which  always  leads  yet  farther,  and  remains 
both  free  from  every  arbitrariness  and  necessary 
and  logical  in  itself." 

*  To  this  passage  Cantor  added  the  note  :  "  I  was  led  to  this  genera- 
tion ten  years  ago  [the  note  was  written  in  May  i88o],  but  when 
exposing  my  theory  of  the  number-conception  I  did  not  refer  to  it." 


.    INTRODUCTION  37 

We  see  that  point-aggregates  of  the  first  kind 
are  characterized  by  the  property  that  P^~^  has  no 
elements,  or,  in  symbols, 

p(-)  =  o^ 

and  also  the  above  example  shows  that  a  point- 
aggregate  of  the  second  kind  need  not  be  every- 
where dense  in  any  part  of  an  interval. 

In  the  first  of  his  papers  of  1882,  Cantor  extended 
the  conceptions  "derivative"  and  "everywhere 
dense "  to  aggregates  situated  in  continua  of  71 
dimensions,  and  also  gave  some  reflexions  on  the 
question  as  to  under  what  circumstances  an  (infinite) 
aggregate  is  well  defined.  These  reflexions,  though 
important  for  the  purpose  of  emphasizing  the 
legitimacy  of  the  process  used  for  defining  P^°°^, 
P^^~\  .  .  .,  are  more  immediately  connected  with 
the  conception  of  "power,"  and  will  thus  be  dealt 
with  later.  The  same  applies  to  the  proof  that  it  is 
possible  to  remove  an  everywhere  dense  aggregate 
from  a  continuum  of  two  or  more  dimensions  in 
such  a  way  that  any  two  points  can  be  connected 
by  continuous  circular  arcs  consisting  of  the  re- 
maining points,  so  that  a  continuous  motion  may 
be  possible  in  a  discontinuous  space.  To  this 
Cantor  added  a  note  stating  that  a  purely  arith- 
metical theory  of  magnitudes  was  now  not  onl}' 
known  to  be  possible,  but  also  already  sketched  out 
in  its  leading  features. 

We  must  now  turn  our  attention  to  the  develop- 
ment   of   the   conceptions  of  "  enumerability "  and 


38  INTRODUCTION 

*' power,"  which  were  gradually  seen  to  have  a 
very  close  connexion  with  the  theory  of  derivatives 
and  the  theory,  arising  from  this  theory,  of  the 
transfinite  numbers. 

In  1873,  Cantor  set  out  from  the  question 
whether  the  linear  continuum  (of  real  numbers) 
could  be  put  in  a  one-one  correspondence  with  the 
aggregate  of  whole  numbers,  and  found  the  rigorous 
proof  that  this  is  not  the  case.  This  proof,  together 
with  a  proof  that  the  totality  of  real  algebraic 
numbers  can  be  put  in  such  a  correspondence,  and 
hence  that  there  exist  transcendental  numbers  in 
every  interval  of  the  number-continuum,  was 
published  in  1874. 

A  real  number  w  which  is  a  root  of  a  non-identical 
equation  of  the  form 

(4)  ^0^^^  +  ^!^""^+  •  •  •   +^«  =  0' 

where  n,  a^,  a^,  .  .  .,  a„  are  integers,  is  called  a  real 
algebraic  number ;  we  may  suppose  n  and  a^  positive, 
^0,  a^,  .  .  .,  a,,  to  have  no  common  divisor,  and  (4) 
to  be  irreducible.      The  positive  whole  number 

N  =  ?2  -  I  -F  I  ^0  I  +  I  ^1  I  +  •  •  •  +  I  ^«  I 
may  be  called  the  'Mieight"  of  w  ;  and  to  each 
positive  integer  correspond  a  finite  number  of  real 
algebraic  numbers  whose  height  is  that  integer. 
Thus  we  can  arrange  the  totality  of  real  algebraic 
numbers  in  a  simply  infinite  series 

by    arranging    the    numbers    corresponding    to    the 


INTRODUCTION  39 

height    N    in    order    of   magnitude,    and    then    the 
various  heights  in  their  order  of  magnitude. 

Suppose,  now,  that  the  totahty  of  the  real 
numbers  in  the  interval  (a  .  .  .  ,8),  where  a</3, 
could  be  arranged  in  the  simply  infinite  series 

(5)  ^1,  u.^,  .  .  .^u,,  .  .  . 

Let  a',  /3'  be  the  two  first  numbers  of  (5),  different 
from  one  another  and  from  a,  /3,  and  such  that 
a  <^'  \  similarly,  let  a\  j^'\  where  a'</3'\  be  the 
first  different  numbers  in  (a  .  .  .  /3'),  and  so  on. 
The  numbers  a,  a" ■>  .  .  .  are  members  of  (5)  whose 
indices  increase  constantly  ;  and  similarly  for  the 
numbers  p\  ^'\  ...  of  decreasing  magnitude. 
Each  of  the  intervals  (a  .  .  .  /3),  (a  .  .  .  /3'), 
{a"  .  .  .  j3"),  .  .  .  includes  all  those  which  follow. 
We  can  then  only  conceive  two  cases  :  either  {a) 
the  number  of  intervals  is  finite  ; — let  the  last  be 
{a^"^  .  .  .  /S^"^)  ;  then,  since  there  is  in  this  interval  at 
most  one  number  of  (5),  we  can  take  in  it  a  number 
rj  which  does  not  belong  to  (5)  ; — or  (d)  there  are 
infinitely  many  intervals.  Then,  since  a,  a,  a\  .  .  . 
increase  constantly  without  increasing  ad  infinituvi^ 
they  have  a  certain  limit  a^°°\  and  similarly  /3,  /3', 
/3'',  .  .  .  decrease  constantly  towards  a  certain 
limit  Z?^"').  If  a^°°)  =  /3(°°)  (which  always  happens 
when  applying  this  method  to  the  system  (o))),  we 
easily  see  that  the  number  ri  =  a^"'^  cannot  be  in  (5).* 
If,   on  the  contrary,  a^°'^<^''~\  every  number  tj  in 

•■  For  if  it  were,  we  would  have  y}  =  up,  p  being  a  determined  index  ; 
but  that  is  not  possible,  for  up  is  not  in  (a^^^  .  .  .  /S^-^^),  whilst  tj,  by 
definition,  is. 


v-^ 


40  INTRODUCTION 

the  interval  {a^"^^  .  .  .  ^(">)  or  equal  to  one  of  its 
ends  fulfils  the  condition  of  not  belonging  to  (5). 

The  property  of  the  totality  of  real  algebraic 
numbers  is  that  the  system  (co)  can  be  put  in  a  one- 
to-one  or  (i,  i)-correspondence  with  the  system 
(i/),  and  hence  results  a  new  proof  of  Liouville's 
theorem  that,  in  every  interval  of  the  real  numbers, 
there  is  an  infinity  of  transcendental  (non-algebraic) 
numbers. 

This  conception  of  (i,  i)-correspondence  between 
aggregates  was  the  fundamental  idea  in  a  memoir 
of  1877,  published  in  1878,  in  which  some  import- 
ant theorems  of  this  kind  of  relation  between  various 
aggregates  were  given  and  suggestions  made  of  a 
classification  of  aggregates  on  this  basis. 

If  two  well-defined  aggregates  can  be  put  into 
such  a  (i,  i)-correspondence  (that  is  to  say,  if, 
element  to  element,  they  can  be  made  to  correspond 
completely  and  uniquely),  they  are  said  to  be 
of  the  same  ' '  power "  {Mdchtigkeit  *)  or  to  be 
' '  equivalent  "  {aequivalent).  When  an  aggregate 
is  finite,  the  notion  of  power  corresponds  to  that  of 
number  {Anzahl),  for  two  such  aggregates  have  the 
same  power  when,  and  only  when,  the  number  of 
their  elements  is  the  same. 

A  part  {Bestandteil ;  any  other  aggregate  whose 
elements  are  also  elements  of  the  original  one)  of  a 
finite  aggregate  has  always  a  power  less  than  that 


*  The  word  "power"  was  borrowed  from  Steiner,  who  used  it  in  a 
quite  special,  but  allied,  sense,  to  express  that  two  figures  can  be  put, 
element  for  element,  in  projective  correspondence. 


INTRODUCTION  41 

of  the  aggregate  itself,  but  this  is  not  always  the 
case  with  infinite  aggregates,* — for  example,  the 
series  of  positive  integers  is  easily  seen  to  have  the 
same  power  as  that  part  of  it  consisting  of  the  even 
integers, — and  hence,  from  the  circumstance  that 
an  infinite  aggregate  M  is  part  of  N  (or  is  equiva- 
lent to  a  part  of  N),  we  can  only  conclude  that  the 
power  of  M  is  less  than  that  of  N  if  we  know  that 
these  powers  are  unequal. 

The  series  of  positive  integers  has,  as  is  easy  to 
show,  the  smallest  infinite  power,  but  the  class  of 
aggregates  with  this  power  is  extraordinarily  rich 
and  extensive,  comprising,  for  example,  Dedekind's 
'*  finite  corpora,"  Cantor's  ''systems  of  points  of 
the  j/th  species,"  all  ;^-ple  series,  and  the  totality  of 
real  (and  also  complex)  algebraic  numbers.  Further, 
we  can  easily  prove  that,  if  M  is  an  aggregate  of 
this  first  infinite  power,  each  infinite  part  of  M  has 
the  same  power  as  M,  and  if  M",  M",  ...  is  a  finite 
or  simply  infinite  series  of  aggregates  of  the  first 
power,  the  aggregate  resulting  from  the  union  of 
these  aggregates  has  also  the  first  power. 

By  the  preceding  memoir,  continuous  aggregates 
have  not  the  first  power,  but  a  greater  one  ;  and 
Cantor  proceeded  to  prove  that  the  analogue,  with 
continua,  of  a  multiple  series — a  continuum  of  many 
dimensions — has  the  same  power  as  a  continuum  of 

"  This  curious  property  of  infinite  aggregates  was  first  noticed  by 
Bernard  Bolzano,  obscurely  stated  ("  .  .  .  two  unequal  lengths  [may 
be  said  to]  contain  the  same  number  of  points")  in  a  paper  of  1864  in 
which  Augustus  De  Morgan  argued  for  a  proper  infinite,  and  was  used 
as  a  definition  of  "infinite"  by  Dedekind  (independently  of  Bolzano 
and  Cantor)  in  1887. 


42  INTRODUCTION 

one  dimension.  Thus  it  appeared  that  the  assump- 
tion of  Riemann,  Helmholtz,  and  others  that  the 
essential  characteristic  of  an  ;2-ply  extended  con- 
tinuous manifold  is  that  its  elements  depend  on  n 
real,  continuous,  independent  variables  (co-ordin- 
ates), in  such  a  way  that  to  each  element  of  the 
manifold  belongs  a  definite  system  of  values  x^^  x^, 
.  .  .,  ;ir„,  and  reciprocally  to  each  admissible  system 
x^^  x^,  .  .  .,  x„  belongs  a  certain  element  of  the 
manifold,  tacitly  supposes  that  the  correspondence 
of  the  elements  and  systems  of  values  is  a  continuous 
one.  *  If  we  let  this  supposition  drop,t  we  can  prove 
that  there  is  a  (i,  i)-correspondence  between  the 
elements  of  the  linear  continuum  and  those  of  a 
n-p\y  extended  continuum. 

This  evidently  follows  from  the  proof  of  the 
theorem:  Let  x^^,  x^,  .  .  .,  x,,  be  real,  independent 
variables,  each  of  which  can  take  any  value  o<x<  i  ; 
then  to  this  system  of  n  variables  can  be  made  to 
correspond  a  variable  /(o</<i)  so  that  to  each 
determined  value  of  t  corresponds  one  system  of 
determined  values  of  x^^,  x,^,  .  .  .,  x,„  and  via  versa. 
To  prove  this,  we  set  out  from  the  known  theorem 
that  every  irrational  number  e  between  o  and  i  can 
be  represented  in  one  manner  by  an  infinite  con- 
tinued fraction  which  may  be  written  : 

(«!,  otg,  .  .  .,   a^,  .  .  .)» 

*  That  is  to  say,  an  infinitely  small  variation  in  position  of  the  element 
imphes  an  infinitely  small  variation  of  the  variables,  and  reciprocally. 

t  In  the  French  translation  only  of  this  memoir  of  Cantor's  is  added 
here:  "and  this  happens  very  often  in  the  works  of  these  authors 
(Riemann  and  Helmholtz)."     Cantor  had  revised  this  translation. 


INTRODUCTION  43 

where  the  a's  are  positive  integers.  There  is  thus 
a  (i,  i)-correspondence  between  the  ^'s  and  the 
various  series  of  a's.  Consider,  now,  n  variables, 
each  of  which  can  take  independently  all  the  ir- 
rational values  (and  each  only  once)  in  the  interval 
(o  .  .  .   I): 

^i  =  (ai,  1,  ai,  2j  '   •   •,  cti,  ^,  .   .  .), 
^a~(^2,  1)  «2,  2>  •••)  a2,  I'j  •••))••   •) 
^/;  =  (a„,  1,  a„,  2>  .  .   •>  a«,  V,  .  .   •)  5 
these    n    irrational    numbers    uniquely  determine  a 
(/^+  i)th  irrational  number  in  (o  .  .  .   i), 

^=(A,  ft,  .  .  .  ft,  .  .  .), 

if  the  relation  between  a  and  /3  : 

(6)   ft.-i)«+A^  =  a^,v*  (m=  I,  2,  .  .  .,  n\  1^=1,  2,  .  .  .CO) 

is  established.  Inversely,  such  a  ^t'  determines 
uniquely  the  series  of  fts  and,  by  (6),  the  series  of 
the  a's,  and  hence,  again  of  the  ^'s.  We  have  only 
to  show,  now,  that  there  can  exist  a  (i,  i)-corre- 
spondence  between  the  irrational  numbers  o<e<\ 
and  the  real  (irrational  and  rational)  numbers 
o<^'<i.  For  this  purpose,  we  remark  that  all 
the  rational  numbers  of  this  interval  can  be  written 
in  the  form  of  a  simply  infinite  series 

^1,    02J    •    •    -J    V-V)    •    •    -t 

*  If  we  arrange  the  n  series  of  o's  in  a  double  series  with  n  rows, 
this  nieans  that  we  are  to  enumerate  the  a's  in  the  order  aj  ^  ^  a.,  ,  ^ 
.  .  .  a^,  J,  Oj  o,  a.,  o,  .  .  .  ,  and  that  the  ^th  term  of  this  series 
is  /3r.      ' 

t  This  is  done  most  simply  as  follows  :  Let  plq  be  a  rational  number 
of  this  interval  in  its  lowest  terms,  and  put  /  +  ^  =  N.     To  each  plq 


44  INTRODUCTION 

Then  in  (o  .  .  .i)  we  take  any  infinite  series  of 
irrational  numbers  tji,  t]2y  -  -  -y  ^vy  •  -  •  (^o^  example, 
tj^=  J  212"),  and  let  /i  take  any  of  the  values  of 
O  .  .  .   i)  except  the  0's  and  >/s,  so  that 

and  we  can  also  write  the  last  formula : 

Now,  if  we  write  a  Oo  d  for  ' '  the  aggregate  of  the  a's 
is  equivalent  to  that  of  the  ^'s,"  and  notice  that  aooa, 
a  c\j  3  and  d  cx)  c  imply  a  c\)  c,  and  that  two  aggre- 
gates of  equivalent  aggregates  of  elements,  where  the 
elements  of  each  latter  aggregate  have,  two  by  two, 
no  common  element,  are  equivalent,  we  remark  that 

and 

X  cso  e. 

A  generalization  of  the  above  theorem  to  the  case 
of  x-^,  x.^,  .  .  .,  x^,  .  .  .  being  a  simply  infinite  series 
(and  thus  that  the  continuum  may  be  of  an  infinity 
of  dimensions  while  remaining  of  the  same  power 
as  the  linear  continuum)  results  from  the  observa- 
tion that,  between  the  double  series   {a^^  „},  where 

^;a  =  («^t,  1  ,     tt/x,  2,    •     .     .  ,    a/x,   «"    •     •     •  )      ^^^'      M  =   I  J     2,    ...  CO 

belongs  a  determined  positive  integral  value  of  N,  and  to  each  such  N 
belong  a  finite  number  of  fractions  />/(/.  Imagine  now  the  numbers //(^ 
arranged  so  that  those  which  belong  to  smaller  values  of  N  precede 
those  which  belong  to  larger  ones,  and  those  for  which  N  has  the  same 
value  are  arranged  the  greater  after  the  smaller. 

*  This  notation  means  :  the  aggregate  of  the  a's  is  the  union  of  those 
of  the  ^'s,  7;»''s,  and  (pyS  ;  and  analogously  for  that  of  the  ^'s. 


INTRODUCTION  45 

and  the  simple  series  {/3;y^},  a  (i,  i)-correspondence 
can  be  established  *  by  putting 

\=fj.-^{fj,^V-  l)(/x  +  j/-2)/2, 

and  the  function  on  the  right  has  the  remarkable 
property  of  representing  all  the  positive  integers, 
and  each  of  them  once  only,  when  /x  and  v  inde- 
pendently take  all  positive  integer  values. 

''And  now  that  we  have  proved,"  concluded 
Cantor,  "for  a  very  rich  and  extensive  field  of 
manifolds,  the  property  of  being  capable  of  corre- 
spondence with  the  points  of  a  continuous  straight 
line  or  with  a  part  of  it  (a  manifold  of  points  con- 
tained in  it),  the  question  arises  .  .  . :  Into  how 
many  and  what  classes  (if  we  say  that  manifolds  of 
the  same  or  different  power  are  grouped  in  the  same 
or  different  classes  respectively)  do  linear  manifolds 
fall  ?  By  a  process  of  induction,  into  the  further 
description  of  which  we  will  not  enter  here,  we  are 
led  to  the  theorem  that  the  number  of  classes  is  two  : 
the  one  containing  all  manifolds  susceptible  of  being 
brought  to  the  form  :  functio  ipsius  v,  where  u  can 
receive  all  positive  integral  values  ;  and  the  other 
containing  all  manifolds  reducible  to  the  ioxxn  functio 
ipsius  ,r,  where  x  can  take  all  the  real  values  in  the 
interval  (o  .  .  .   i). " 

In  the  paper  of  1879  already  referred  to,  Cantor 

*   Enumerate  the  double  series   |a^^  ^,\   diagonally,   that  is  to  say, 
in  the  order 


The   term   of   this   series   whose   index   is   (/i,   v)    is   the   Ath,    where 
A=I+2  +  3+    .    .    .+()U  +  l/-2)  +  /i  =  (/A  +  l'-2)(;i  +  |/-l)/2  +  /x. 


46  INTRODUCTION 

considered  the  classification  of  aggregates  *  both 
according  to  the  properties  of  their  derivatives  and 
according  to  their  powers.  After  some  repetitions, 
a  rather  simpler  proof  of  the  theorem  that  the  con- 
tinuum is  not  of  the  first  power  was  given.  But, 
though  no  essentially  new  results  on  power  were 
published  until  late  in  1882,  we  must  refer  to  the 
discussion  (1882)  of  what  is  meant  by  a  "well- 
defined  "  aggregate. 

The  conception  of  power  f  which  contains,  as  a 
particular  case,  the  notion  of  whole  number  may, 
said  Cantor,  be  considered  as  an  attribute  of  every 
"well-defined"  aggregate,  whatever  conceivable 
nature  its  elements  may  have.  ' '  An  aggregate  of 
elements  belonging  to  any  sphere  of  thought  is  said 
to  be  '  well  defined '  when,  in  consequence  of  its 
definition  and  of  the  logical  principle  of  the  excluded 
middle,  it  must  be  considered  as  intrinsically  deter- 
mined whether  any  object  belonging  to  this  sphere 
belongs  to  the  aggregate  or  not,  and,  secondly, 
whether  two  objects  belonging  to  the  aggregate 
are  equal  or  not,  in  spite  of  formal  differences 
in  the  manner  in  which  they  are  given.  In  fact, 
we  cannot,  in  general,  effect  in  a  sure  and  precise 
manner  these  determinations  with  the  means  at  our 
disposal  ;  but  here  it  is  only  a  question  of  intrinsic 
determination,   from  which  an    actual    or    extrinsic 

"  Linear  aggregates  alone  were  considered,  since  all  the  powers  of 
the  continua  of  various  dimensions  are  to  be  found  in  them. 

t  "That  foundation  of  the  theory  of  magnitudes  which  we  may 
consider  to  be  the  most  general  genuine  moment  in  the  case  of 
manifolds." 


INTRODUCTION  47 

determination  is  to  be  developed  by  perfecting  the 
auxiliary  means."  Thus,  we  can,  without  any 
doubt,  conceive  it  to  be  intrinsically  determined 
whether  a  number  chosen  at  will  is  algebraic  or 
not  ;  and  yet  it  was  only  proved  in  1874  that  e  is 
transcendental,  and  the  problem  with  regard  to  tt 
was  unsolved  when  Cantor  wrote  in  1882.* 

In  this  paper  was  first  used  the  word  **  enumer- 
able "  to  describe  an  aggregate  which  could  be  put 
in  a  (i,  i)-correspondence  with  the  aggregate  of 
the  positive  integers  and  is  consequently  of  the  first 
(infinite)  power  ;  and  here  also  was  the  important 
theorem  :  In  a  ;/-dimensional  space  (A)  are  defined 
an  infinity  of  (arbitrarily  small)  continua  of  11 
dimensions  f  {a)  separated  from  one  another  and 
most  meeting  at  their  boundaries  ;  the  aggregate  of 
the  rt:'s  is  enumerable. 

For  refer  A  by  means  of  reciprocal  radii  vectores 
to  an  ;^-ply  extended  figure  B  within  a  («+  i)- 
dimensional  infinite  space  /\',  and  let  the  points  of 
B  have  the  constant  distance  i  from  a  fixed  point 
of  A'.  To  every  a  corresponds  a  /^-dimensional 
part  <^  of  B  with  a  definite  content,  and  the  ^'s  are 
enumerable,  for  the  number  of  b's  greater  in  con- 
tent than  an  arbitrarily  small  number  y  is  finite,  for 
their  sum  is  less  than  2'V  +  (the  content  of  B).  § 

*  Lindemann  afterwards  proved  that  it  is  transcendental.  In  this 
passage,  Cantor  seemed  to  agree  with  Dedekind. 

t  With  every  a  the  points  of  its  boundary  are  considered  as  belong- 
ing to  it. 

:;:  In  the  French  translation  (1883)  of  Cantor's  memoir,  this  num])er 
was  corrected  to  Qw(m  I  lyg/F^fyr-MH/g). 

§  When  n=i,  the  theorem  is  that  every  aggregate  of  intervals  on  a 

2^     -i.  cf,M,.tt..A..n    t^Zl     ^.fT'^P, 


r(^-r) 


48  INTRODUCTION 

Finally,  Cantor  made  the  interesting  remark  that, 
if  we  remove  from  an  ;^-dimensional  continuum  any 
enumerable  and  everywhere-dense  aggregate,  the 
remainder  (21),  if  ^>2,  does  not  cease  to  be  con- 
tinuously connected,  in  the  sense  that  any  two 
points  N,  N'  of  51  can  be  connected  by  a  continuous 
line  composed  of  circular  arcs  all  of  whose  points 
belong  to  51. 

VI 

An  application  of  Cantor's  conception  of  enumera- 
bility  was  given  by  a  simpler  method  of  condensation 
of  singularities,  the  construction  of  functions  having  " 
a  given  singularity,  such  as  a  discontinuity,  at  an 
enumerable  and  everywhere-dense  aggregate  in  a 
given  real  interval.  This  was  suggested  by  Weier- 
strass,  and  published  by  Cantor,  with  Weierstrass's 
examples,  in  1882.*  The  method  may  be  thus 
indicated  :  Let  ^p{x)  be  a  given  function  with  the 
single    singularity  x—O,    and   (w^,)   any   enumerable 

aggregate  ;  put 

00 

v  =  l 

where  the  ^^'s  are  so  chosen  that  the  series  and 
those  derived  from  it  in  the  particular  cases  treated 
converge    unconditionally    and    uniformly.       Then 

(finite  or  infinite)  straight  line  which  at  most  meet  at  their  ends  is 
enumerable.  The  end-points  are  consequently  enumerable,  but  not 
always  the  derivative  of  this  aggregate  of  end-points. 

*  Inaletter  tome  of  29th  March  1905,  Professor  Cantor  said  :  "Atthe 
conception  of  enumerability,  of  which  he  [Weierstrass]  heard  from  me  at 
Berlin  in  the  Christmas  holidays  of  1873,  l^e  was  at  first  quite  amazed, 
but  one  or  two  days  passed  over,  [and]  it  became  his  own  and  helped  him 
to  an  unexpected  development  of  his  wonderful  theory  of  functions." 


INTRODUCTION  49 

f{x)  has  at  all  points  A'  =  a)^  the  same  kind  of  singu- 
larity as  (^{x)  at;ir=o,  and  at  other  points  behaves, 
in  general,  regularly.  The  singularity  at  x  =  {^^  is 
due  exclusively  to  the  one  term  of  the  series  in 
which  j/  =  /u  ;  the  aggregate  (co^)  may  be  any  enumer- 
able aggregate  and  not  only,  as  in  Hankel's  method, 
the  aggregate  of  the  rational  numbers,  and  the 
superfluous  and  complicating  oscillations  produced 
by  the' occurrence  of  the  sine  in  Hankel's  functions 
is  avoided. 

The  fourth  (1882)  of  Cantor's  papers  "  (Jeber 
unendliche,  lineare  Punktmannichfaltigkeiten  "  con- 
tained six  theorems  on  enumerable  point-aggregatfes. 
H^an  aggregate  O  (in  a  continuum  of  n  dimensions) 
is  such  tha't  none  of  its  points  is  a  limit-point,*  it  is 
said  to  be  "  isolated."  Then,  round  every  point  of 
O  a  sphere  can  be  drawn  which  contains  no  other 
point  of  Q,  and  hence,  by  the  above  theorem  on 
the  enumerability  of  the' aggregate  of  these  spheres, 
is  enumerable. 

Secondly,  if  P'  is  enumerable,  P  is.      P'or  let 

^(P,  P')^R,  P-R^O;t 

then  O  is  isolated  and  therefore  enumerable,  and  R 
is  also  enumerable,  since  R  is  contained  in  P' ;  so 
P  is  enumerable. 

The  next  three  theorems   state    that,   if   P<''\   or 

*  Cantor  expressed  this  X'(Q>  Q')  — O-  Q-  Dedekind's  Essays  on 
Nuf/iber,  p.  48. 

t  If  an  aggregate  B  is  contained  in  A,  and  E  is  the  aggregate  left 
when  B  is  taken  from  A,  we  write 

E  =  A-B. 


50  INTRODUCTION 

P^«>,  where  a  is  any  one  of  the  ' '  definitely  defined 
symbols  of  infinity  {bestimmt  definirte  Unendlich- 
keitssymbole),''  is  enumerable,  then  P  is. 

If  the  aggregates  1\,  V^,  .  .  .  have,  two  by  two, 
no  common  point,  for  the  aggregate  P  formed  by 
the  union  of  these  (the  ''  Vereinigungsmenge'')  Cantor 
now  used  the  notation 

P^P,  +  P2+.  .. 
Now,  we  have  the  following  identity 
P'^(P'-P")  +  (P"-P"0+  .  •  •  +(P(''-i)-P(''))+P^^^ 
and  thus,  since 

p/_p//    Y" —V"       .  .    p(''-i)  — p('') 

are  all  isolated  and  therefore  enumerable,  if  P^'^^  is 
enumerable,  then  P'  is  also. 

Now,  suppose  that  P(~>  exists  ;  then,  if  any  par- 
ticular point  of  P'  does  not  belong  to  P^°°),  there  is  a 
first  one  among  the  derivatives  of  finite  order,  P^"^), 
to  which  it  does  not  belong,  and  consequently  P^^-^) 
contains  it  as  an  isolated  point.  Thus  we  can  write 
P'^(P^_p-)  +  (p-_p-)+  .  .  .    +(p(-i)-PM) 

+  .  .  .    +P(~); 

and  consequently,  since  an  enumerable  aggregate  of 
enumerable  aggregates  is  an  enumerable  aggregate 
of  the  elements  of  the  latter,  and  P^"^^  is  enumerable, 
then  P'  is  also.  This  can  evidently  be  extended  to 
P^*^),  if  it  exists,  provided  that  the  aggregate  of  all 
the  derivatives  from  P'  to  P^*^  is  enumerable. 

The    considerations    which    arise    from    the    last 


INTRODUCTION  51 

observation  appear  to  me  to  have  constituted  the 
final  reason  for  considering  these  definitely  infinite 
indices  independently  *  on  account  of  their  con- 
nexion with  the  conception  of  ''power,"  which 
Cantor  had  always  regarded  as  the  most  funda- 
mental one  in  the  whole  theory  of  aggregates. 
The  series  of  the  indices  found,  namely,  is  such 
that,  up  to  any  point  (infinity  or  beyond),  the 
aggregate  of  them  is  always  enumerable,  and  yet 
a  process  exactly  analogous  to  that  used  in  the 
proof  that  the  continuum  is  not  enumerable  leads 
to  the  result  that  the  aggregate  of  all  the  indices 
such  that,  if  a  is  any  index,  the  aggregate  of  all  the 
indices  preceding  a  is  enumerable,  is  not  enumer- 
able, but  is,  just  as  the  power  of  the  series  of 
positive  integers  is  the  next  higher  one  to  all  finite 
ones,  the  next  greater  infinite  power  to  the  first. 
And  we  can  again  imagine  a  new  index  which  is  the 
first  after  all  those  defined,  just  as  after  all  the  finite 
ones.  We  shall  see  these  thoughts  published  by 
Cantor  at  the  end  of  1882. 

It  remains  to  mention  the  sixth  theorem,  in 
which  Cantor  proved  that,  if  P'  is  enumerable,  P 
has  the  property,  which  is  essential  in  the  theory 
of  integration,  of  being  "discrete,"  as  Harnack 
called  it,  "  integrable,"  as  P.  du  Bois-Reymond  did, 
'' unextended,"  or,  as  it  is  now  generally  called, 
"  content-less." 

*  When  considered  independently  of  P,  these  indices  form  a  series 
beginning  with  the  finite  numbers,  but  extending  beyond  them;  so 
that  it  suggests  itself  that  those  other  indices  be  considered  as  infinite 
(or  transhnite)  tnanbers. 


52  INTRODUCTION 

VII 

VVe  have  thus  seen  the  importance  of  Cantor's 
"definitely  defined  symbols  of  infinity"  in  the 
theorem  that  if  P^"^  vanishes,  P',  and  therefore  P,  is 
enumerable.  This  theorem  may,  as  we  can  easily 
see  by  what  precedes,  be  inverted  as  follows  :  If 
P'  is  enumerable,  there  is  an  index  a  such  that  P^"^ 
vanishes.  By  defining  these  indices  in  an  inde- 
pendent manner  as  real,  and  in  general  transfinite, 
integers.  Cantor  was  enabled  to  form  a  conception 
of  the  enumeral  *  {Anza/il)  of  certain  infinite  series, 
and  such  series  gave  a  means  of  defining  a  series  of 
ascending  infinite  "powers."  The  conceptions  of 
"enumeral"  and  "power"  coincided  in  the  case  of 
finite  aggregates,  but  diverged  in  the  case  of  infinite 
aggregates  ;  but  this  extension  of  the  conception  of 
enumeral  served,  in  the  way  just  mentioned,  to 
develop  and  make  precise  the  conception  of  power 
used  often  already. 

Thus,  from  the  new  point  of  view  gained,  we  get 
new  insight  into  the  theory  of  finite  number  ;  as 
Cantor  put  it  :  "The  conception  of  number  which, 
in  finito,  has  only  the  background  of  enumeral, 
splits,  in  a  manner  of  speaking,  when  we  raise  our- 
selves to  the  infinite,  into  the  two  conceptions  of 
power  .  .  .  and  enumeral .  .  .  ;  and,  when  I  again 
descend  to  the  finite,  I  see  just  as  clearly  and 
beautifully  how  these  two  conceptions  again  unite 
to  form  that  of  the  finite  integer." 

*  I  have  invented  this  woid  to  translate  "  Anzahl,"  to  avoid  confusion 
with  the  word  "  number"  [Zahl). 


INTRODUCTION  53 

The  significance  of  this  distinction  for  the  theor}' 
of  all  (finite  and  infinite)  arithmetic  appears  in 
Cantor's  own  work  *  and,  above  all,  in  the  later 
work  of  Russell. 

Without  this  extension  of  the  conception  of 
number  to  the  definitely  infinite  numbers,  said 
Cantor,  "  it  would  hardly  be  possible  for  me  to 
make  without  constraint  the  least  step  forwards  in 
the  theory  of  aggregates,"  and,  although  "I  was 
led  to  them  [these  numbers]  many  years  ago, 
without  arriving  at  a  clear  consciousness  that  1 
possessed  in  them  concrete  numbers  of  real  signi- 
ficance," yet  "  I  was  logically  forced,  almost  against 
my  will,  because  in  opposition  to  traditions  which 
had  become  valued  by  me  in  the  course  of  scientific 
researches  extending  over  many  years,  to  the 
thought  of  considering  the  infinitely  great,  not 
merely  in  the  form  of  the  unlimitedly  increasing, 
and  in  the  form,  closely  connected  with  this,  of 
convergent  infinite  series,  but  also  to  fix  it  mathe- 
matically by  numbers  in  the  definite  form  of  a 
'completed  infinite.'  I  do  not  believe,  then,  that 
any  reasons  can  be  urged  against  it  which  I  am 
unable  to  combat." 

The  indices  of  the  series  of  the  derivatives  can 
be    conceived    as    the    series     of     finite     numbers 

I,    2, ,   followed  by  a  series  of  tra7isfiuite 

numbers  of  which  the  first  had  been  denoted  b}-  the 
symbol  "00."     Thus,  although  there  is  no  greatest 

*  C/:,  for  example,  pp.  1 1 3,  1 58-159  of  the  translations  of  Cantor's 
memoirs  of  1895  and  1897  given  below. 


U' 


54  INTRODUCTION 

finite  number,  or,  in  other  words,  the  supposition 
that  there  is  a  greatest  finite  number  leads  to  con- 
tradiction, there  is  no  contradiction  involved  in 
postulating  a  new,  non-finite,  number  which  is  to  be 
the  first  after  all  the  finite  numbers.  This  is  the 
method  adopted  by  Cantor  *  to  define  his  numbers 
independently  of  the  theory  of  derivatives  ;  we  shall 
see  how  Cantor  met  any  possible  objections  to  this 
system  of  postulation. 

Let  us  now  briefly  consider  again  the  meaning  of 
the  word  '■^  MannichfaltigkeitsleJire^'"  ^  which  is 
usually  translated  as  "  theory  of  aggregates."  In  a 
note  to  the  Gyundlagen^  Cantor  remarked  that  he 
meant  by  this  word  ' '  a  doctrine  embracing  very 
much,  which  hitherto  1  have  attempted  to  develop 
only  in  the  special  form  of  an  arithmetical  or 
geometrical  theory  of  aggregates  {Mengenlehre). 
By  a  manifold  or  aggregate  I  understand  generally 
any  multiplicity  which  can  be  thought  of  as  one 
(jedes  Viele,  welches  sick  als  Eines  denken  lasst),  that 
is  to  say,  any  totality  of  definite  elements  which 
can  be  bound  up  into  a  whole  by  means  of  a  law." 

*  "  Ueber  unendliche,  lineare  Punktmannichfaltigkeiten.  V." 
[December  1882],  Math.  Ann.,  vol.  xxi,  1883,  pp.  545-591  ;  reprinted, 
with  an  added  preface,  with  the  title  :  Griindlagen  einer  allgevieinen 
Mamiichfaltigkeitslehre.  Ein  uiathematisch-philosophischer  Versiich  in 
der  Lehre  des  Unendiichen,'Lit\\:>z\g,  1883  (page  n  of  the  Grnndlagenh 
page  w  +  544  of  the  article  in  the  Math.  Ann.).  This  separate  publica- 
tion, with  a  title  corresponding  more  nearly  to  its  contents,  was  made 
"  since  it  carries  the  subject  in  many  respects  much  farther  and  thus  is, 
for  the  most  part,  independent  of  the  earlier  essays"  (Preface).  In 
Acta  Math.,  ii,  pp.  381-408,  part  of  the  Grundlagen  was  translated 
into  French.  • 

t  Or  •''  Manntgfaitigkeitslehre," or,  more  usually, "  Mengenlehre  "  ;  in 
French,  ^^  th^orte  des  ensembles."  The  English  "  theory  of  manifolds" 
has  not  come  into  general  usage. 


INTRODUCTION  55 

This  character  of  unity  was  repeatedly  emphasized 
by  Cantor,  as  we  shall  see  later. 

The  above  quotations  about  the  slow  and  sure 
way  in  which  the  transfinite  numbers  forced  them- 
selves on  the  mind  of  Cantor  and  about  Cantor's 
philosophical  and  mathematical  traditions  are  taken 
from  the  Grundlagen.  Both  here  and  in  Cantor's 
later  works  we  constantly  come  across  discussions 
of  opinions  on  infinity  held  by  mathematicians  and 
philosophers  of  all  times,  and  besides  such  names  as 
Aristotle,  Descartes,  Spinoza,  Hobbes,  Berkeley, 
Locke,  Leibniz,  Bolzano,  and  many  others,  we  find 
evidence  of  deep  erudition  and  painstaking  search 
after  new  views  on  infinity  to  analyze.  Cantor  has 
devoted  many  pages  to  the  Schoolmen  and  the 
Fathers  of  the  Church. 

The  Grundlagen  begins  by  drawing  a  distinction 
between  two  meanings  which  the  word  "infinity" 
may  have  in  mathematics.  The  mathematical 
infinite,  says  Cantor,  appears  in  two  forms  :  Firstly, 
as  an  improper  infinite  {Uneigentlich-Unendliches), 
a  magnitude  which  either  increases  above  all  limits 
or  decreases  to  an  arbitrary  smallness,  but  always 
remains  finite  ;  so  that  it  may  be  called  a  variable 
finite.  Secondly,  as  a  definite,  a  proper  infinite 
{Eigentlich-Unendliches),  represented  by  certain 
conceptions  in  geometry,  and,  in  the  theory  of 
functions,  by  the  point  infinity  of  the  complex  plane. 
In  the  last  case  we  have  a  single,  definite  point, 
and  the  behaviour  of  (analytic)  functions  about  this 
point  is  examined  in  exactly  the  same  way  as  it   is 


56  INTRODUCTION 

about  any  other  point.*  Cantor's  infinite  real 
integers  are  also  properly  infinite,  and,  to  emphasize 
this,  the  old  symbol  '*  od  ,"  which  was  and  is  used  also 
for  the  improper  infinite,  was  here  replaced  by  ''w." 
To  define  his  new  numbers,  Cantor  employed  the 
following  considerations.  The  series  of  the  real 
positive  integers, 

(I)  I,  2,  3,    .  .  .,     V,    .  .  ., 

arises  from  the  repeated  positing  and  uniting  of 
units  which  are  presupposed  and  regarded  as  equal  ; 
the  number  v  is  the  expression  both  for  a  definite 
finite  enumeral  of  such  successive  positings  and  for 
the  uniting  of  the  posited  units  into  a  whole.  Thus 
the  formation  of  the  finite  real  integers  rests  on  the 
principle  of  the  addition  of  a  unit  to  a  number 
which  has  already  been  formed  ;  Cantor  called  this 
moment  th^fiist principle  of  gene^'ation  {Erzeugungs- 
princip).  The  enumeral  of  the  number  of  the  class 
(I)  so  formed  is  infinite,  and  there  is  no  greatest 
among  them.  Thus,  although  it  would  be  contra- 
dictory to  speak  of  a  greatest  number  of  the  class  (I), 
there  is,  on  the  other  hand,  nothing  objectionable 
in  imagining  a  new  number,  w,  which  is  to  express 
that  the  whole  collection  (I)  is  given  by  its  law  in 
its  natural  order  of  succession  (in  the  same  way  as 
V  is  the  expression  that  a  certain  finite  enumeral  of 
units  is  united  to  a  whole),  f     By  allowing  further 

*  "The  behaviour  of  the  function  in  the  neighbourhood  of  the 
infinitely  distant  point  shows  exactly  the  same  occurrences  as  in  that 
of  any  other  point  lying  iii  finito,  so  that  hence  it  is  completely  justified 
to  think  of  the  infinite,  in  this  case,  as  situated  in  a  point." 

t  "  It  is  even  permissible  to  think  of  the  newly  and  created  number 


INTRODUCTION  57 

positings  of   unity    to    follow    the    positing    of   the 
number   o),   we   obtain    with    the    help   of   the    first 
principle  of  generation  the  further  numbers  : 
w+  I,     (0+2,     .  .  .,     a)  +  j^,     ... 

Since  again  here  we  come  to  no  greatest  number,  we 
imagine  a  new  one,  which  we  may  call  2a),  and  which 
is  to  be  the  first  which  follows  all  the  numbers  v  and 
o)  + 1^  hitherto  formed.  Applying  the  first  principle  re- 
peatedly to  the  number  2co,  we  come  to  the  numbers  : 

g^^      2a)  +  I,      2a) +2,      .   .   .,      2a) +i',      ... 

The  logical  function  which  has  given  us  the 
numbers  a)  and  2a)  is  obviously  different  from  the 
first  principle  ;  Cantor  called  it  the  second  principle 
of  generation  of  real  integers,  and  defined  it  more 
closely  as  follows  :  If  there  is  defined  any  definite 
succession  of  real  integers,  of  which  there  is  no 
greatest,  on  the  basis  of  this  second  principle  a  new 
number  is  created,  which  is  defined  as  the  next  greater 
number  to  them  all. 

By   the  combined    application  of  both  principles 
we  get,  successively,  the  numbers  : 
30),  3a)  +  I ,  .  .  . ,  3a)  + 1/,  .  .  . ,  .  .  . ,  /xa),  .  .  . ,  //a)  +  j/,  .  .  . 

w  as  the  li7nit  to  which  the  numbers  v  strive,  if  by  that  nothing  else  is 
understood  than  that  «  is  to  be  the  first  integer  which  follows  all  the 
numbers  v,  that  is  to  say,  is  to  be  called  greater  than  every  »/."  Cf. 
the  next  section. 

If  we  do  not  know  the  reasons  in  the  theory  of  derivatives  which 
prompted  the  introduction  of  a>,  but  only  the  grounds  stated  in  the  text 
for  this  introduction,  it  naturally  seems  rather  arbitrary  (not  apparently^ 
useful)  to  create  &j  because  of  the  mere  fact  that  it  can  apparently  be 
defined  in  a  manner  free  from  contradiction.  Thus,  Cantor  discussed 
(see  below)  such  introductions  or  creations,  found  in  them  the  dis- 
tinguishing mark  of  pure  mathematics,  and  justified  them  on  historical 
grounds  (on  logical  grounds  they  perhaps  seem  "to  need  no  justification). 


58  INTRODUCTION 

and,  since  no  number  fxw-\-v  is  greatest,  we  create 
a  new  next  number  to  all  these,  which  may  be 
denoted  by  oo^.  To  this  follow,  in  succession, 
numbers  : 

Xo)'^  +  fX(Jd  +  V, 

and  further,  we  come  to  numbers  of  the  form 

and  the  second  principle  then  requires  a  new  number, 
which  may  conveniently  be  denoted  by 

And  so  on  indefinitely. 

Now,  it  is  seen  without  difficulty  that  the 
aggregate  of  all  the  numbers  preceding  any  of  the 
infinite  numbers  and  hitherto  defined  is  of  the 
power  of  the  first  number-class  (I).  Thus,  all  the 
numbers  preceding  w*^  are  contained  in  the  formula  : 

where  /x,  i/q,  i/^,  .  .  . ,  i/^  have  to  take  all  finite, 
positive,  integral  values  including  zero  and  exclud- 
ing the  combination  v^  =  v^=  .  .  .  =v^=zO.  As  is 
well  known,  this  aggregate  can  be  brought  into  the 
form  of  a  simply  infinite  series,  and  has,  therefore, 
the  power  of  (I).  Since,  further,  every  sequence 
(itself  of  the  first  power)  of  aggregates,  each  of 
which  has  the  first  power,  gives  an  aggregate  of  the 
first  power,  it  is  clear  that  we  obtain,  by  the  con- 
tinuation of  our  sequence  in  the  above  way,  only 
such  numbers  with  which  this  condition  is  fulfilled. 


INTRODUCTION  59 

Cantor  defined  the  totality  of  all  the  numbers  a 
formed  by  the  help  of  the  two  principles 

(II)       o),  CD+I,  .  .  .,  j.Qfo'^H-i/iw'^-H  .  .  .  +j/^_r+>, 


such  that  all  the  numbers,  from  i  on,  preceding  a 
form  an  aggregate  of  the  power  of  the  first  number- 
class  (I),  as  the  ^^  second  number-class  (II)."  The 
power  of  (II)  is  different  from  that  of  (I),  and  is, 
indeed,  the  next  higher  power,  so  that  no  other 
power  lies  between  them.  Accordingly,  the  second 
principle  demands  the  creation  of  a  new  number  (Q) 
which  follows  all  the  numbers  of  (II)  and  is  the 
first  of  the  third  number-class  (III),  and  so  on.* 

Thus,  in  spite  of  first  appearances,  a  certain 
completion  can  be  given  to  the  successive  formation 
of  the  numbers  of  (II)  which  is  similar  to  that 
limitation  present  with  (I).  There  we  only  used 
the  first  principle,  and  so  it  was  impossible  to 
emerge  from  the  series  (I)  ;  but  the  second  principle 
must  lead  not  only  over  (II),  but  show  itself  indeed 
as  a  means,  which,  in  combination  with  the  first 
principle,  gives  the  capacity  to  break  through  every 
limit  in  the  formation  of  real  integers.  The  above- 
mentioned  requirement,  that  all  the  numbers  to 
be  next  formed  should  be  such  that  the  aggregate 

^'  It  is  particularly  to  be  noticed  that  the  second  principle  will  lake 
us  beyond  any  class,  and  is  not  merely  adequate  to  form  numbers  which 
are  the  limit-numbers  of  some  enumerable  series  (so  that  a  "third 
principle "  is  required  to  form  fl).  The  first  and  second  principles 
together  form  all  the  numbers  considered,  while  the  "principle  of 
limitation"  enables  us  to  define  the  various  number-classes,  of  un- 
brokenly  ascending  powers  in  the  series  of  these  numbers. 


6o  INTRODUCTION 

of  numbers  preceding  each  one  should  be  of  a  certain 
power,  was  called  by  Cantor  the  third  or  limitation- 
principle  {Hemmungs-  oder  Beschrdnkungsprincip)* 
and  which  acts  in  such  a  manner  that  the  class  (II) 
defined  with  its  aid  can  be  shown  to  have  a  higher 
power  than  (I)  and  indeed  the  next  higher  power  to 
it.  In  fact,  the  two  first  principles  together  define 
an  absolutely  infinite  sequence  of  integers,  while  the 
third  principle  lays  successively  certain  limits  on 
this  process,  so  that  we  obtain  natural  segments 
{Absclmitte),  called  number-classes,  in  this  sequence. 

Cantor's  older  (1873,  1878)  conception  of  the 
''power"  of  an  aggregate  was,  by  this,  developed 
and  given  precision.  With  finite  aggregates  the 
power  coincides  with  the  enumeral  of  the  elements, 
for  such  aggregates  have  the  same  enumeral  of 
elements  in  every  order.  With  infinite  aggregates, 
on  the  other  hand,  the  transfinite  numbers  afford  a 
means  of  defining  the  enumeral  of  an  aggregate,  if 
it  be  "well  ordered,"  and  the  enumeral  of  such  an 
aggregate  of  given  power  varies,  in  general,  with 
the  order  given  to  the  elements.  The  smallest 
infinite  power  is  evidently  that  of  (I),  and,  now  for 
the  first  time,  the  successive  higher  powers  also 
receive  natural  and  simple  definitions  ;  in  fact,  the 
power  of  the  yth  number  class  is  the  yth. 

By  a   "well-ordered"  aggregate,!  Cantor  under- 

*  "This  principle  (or  requirement,  or  condition)  circuaiscribes 
{limits)  each  number-class." 

t  The  origin  of  this  conception  can  easily  be  seen  to  be  the  defining 
of  such  aggregates  as  can  be  "enumerated"  (using  the  word  in  the 
wider  sense  of  Cantor,  given  below)  by  the  transtinite  numbers.  In 
fact,  the  above  definition  of  a  wejl-ordered  aggregate  simply  indicates 


INTRODUCTION  61 

stood  any  well-defined  aggregate  whose  elements 
have  a  given  definite  succession  such  that  there  is 
2.  first  element,  a  definite  element  follows  every  one 
(if  it  is  not  the  last),  and  to  any  finite  or  infinite 
aggregate  a  definite  element  belongs  which  is  the 
next  following  element  in  the  succession  to  them 
all  (unless  there  are  no  following  elements  in  the 
succession).  Two  well-ordered  aggregates  are,  now, 
of  the  same  enumeral  (with  reference  to  the  orders 
of  succession  of  their  elements  previously  given  for 
them)  if  a  one-to-one  correspondence  is  possible 
between  them  such  that,  if  E  and  F  are  any  two 
different  elements  of  the  one,  and  E'  and  F'  the 
corresponding  elements  (consequently  different)  of 
the  other,  if  E  precedes  or  follows  F,  then  E' 
respectively  precedes  or  follows  F'.  This  ordinal 
correspondence  is  evidently  quite  determinate,  if  it 
is  possible  at  all,  and  since  there  is,  in  the  extended 
number-series,  one  and  only  one  number  a  such  that 
its  preceding  numbers  (from  i  on)  in  the  natural 
succession  have  the  same  enumeral,  we  must  put  a 
for  the  enumeral  of  both  well-ordered  aggregates,  if 
a  is  infinite,  or  a—  I  if  a  is  finite. 

The  essential  difference  between  finite  and  infinite 
aggregates  is,  now,  seen  to  be  that  a  finite  aggregate 
has  the  same  enumeral  whatever  the  succession  of 

the  construction  of  any  aggregate  of  the  class  required  when  the  first 
two  principles  are  used,  but  lo  generate  elements,  not  numbers. 

An  important  property  of  a  well-ordered  aggregate,— indeed,  a 
characteristic  property, — is  that  any  series  of  terms  in  it,  ^j ,  ao ,  .  .  ., 
«^  ,  .  .  .,  where  «^+i  precedes  av  ,  must  be  finite.  Even  if  the  well- 
ordered  aggregate  in  question  is  infinite,  such  a  series  as  that  described 
can  never  be  infinite. 


62  INTRODUCTION 

the  elements  may  be,  but  an  infinite  aggregate  has, 
in  general,  different  enumerals  under  these  circum- 
stances. However,  there  is  a  certain  connexion 
between  enumeral  and  power — an  attribute  of  the 
aggregate  which  is  independent  of  the  order  of  the 
elements.  Thus,  the  enumeral  of  any  well-ordered 
aggregate  of  the  first  power  is  a  definite  number  of 
the  second  class,  and  every  aggregate  of  the  first 
power  can  always  be  put  in  such  an  order  that  its 
enumeral  is  any  prescribed  number  of  the  second 
class.  Cantor  expressed  this  by  extending  the 
meaning  of  the  word  "enumerable"  and  saying: 
Every  aggregate  of  the  power  of  the  first  class  is 
enumerable  by  numbers  of  the  second  class  and  only 
by  these,  and  the  aggregate  can  always  be  so 
ordered  that  it  is  enumerated  by  any  prescribed 
number  of  the  second  class  ;  and  analogously  for 
the  higher  classes. 

From    his    above    remarks    on    the    "absolute"* 

*  Cantor  said  "that,  in  the  successive  formation  of  number-classes, 
we  can  always  go  farther,  and  never  reach  a  limit  that  cannot  be  sur- 
passed,— so  that  we  never  reach  an  even  approximate  comprehension 
{Erfasseti)  of  the  Absolute,— I  cannot  doubt.  The  Absolute  can 
only  be  recognized  {anerkannt),  but  never  apprehended  {erkannt), 
even  approximately.  For  just  as  inside  the  first  number-class,  at  any 
finite  number,  however  great,  we  always  have  the  same  '  power '  of 
greater  finite  numbers  before  us,  there  follows  any  transfinite  number 
of  any  one  of  the  higher  number-classes  an  aggregate  of  numbers  and 
classes  which  has  not  in  the  least  lost  in  '  power  '  in  comparison  with  the 
whole  absolutely  infinite  aggregate  of  numbers,  from  i  on.  The  state 
of  things  is  like  that  described  by  Albrecht  von  Haller  :  '  ich  zieh' 
sie  ab  [die  ungeheure  ZahlJ  und  I^u  [die  Ewigkeit]  liegst  ganz  vor  mir.' 
The  absolutely  infinite  sequence  of  numbers  thus  seems  to  me  to  be,  in 
a  certain  sense,  a  suitable  symbol  of  the  Absolute  ;  whereas  the  infinity 
of  (I),  which  has  hitherto  served  for  that  purpose,  appears  to  me,  just 
because  1  hold  it  to  be  an  idea  (not  presentation)  that  can  be  appre- 
hended as  a  vanishing  nothing  in  comparison  with  the  former.  It  also 
seems  to  me  remarkable  that  every  number-class — and  therefore  every 


INTRODUCTION  63 

infinity  of  the  series  of  ordinal  numbers  and  that  of 
powers,  it  was  to  be  expected  that  Cantor  would 
derive  the  idea  that  any  aggregate  could  be  arranged 
in  a  well-ordered  series,  and  this  he  stated  with  a 
promise  to  return  to  the  subject  later.* 

The  addition  and  multiplication  of  the  transfinite 
(including  the  finite)  numbers  was  thus  defined  by 
Cantor.  Let  M  and  M^  be  well-ordered  aggregates 
of  enumerals  a  and  ^,  the  aggregate  which  arises 
when  first  M  is  posited  and  then  M^^,  following  it, 
and  the  two  are  united  is  denoted  M  -f  M^  and  its 
enumeral  is  defined  to  be  a-\- ^.  Evidently,  if  a 
and  /3  are  not  both  finite,  a  +  /3  is,  in  general, 
different  from  /^  +  a.  It  is  easy  to  extend  the  con- 
cept of  sum  to  a  finite  or  transfinite  aggregate  of 
summands  in  a  definite  order,  and  the  associative 
law  remains  valid.      Thus,  in  particular, 

a  +  (/3  +  y)-(a+iS)  +  y. 

If  we  take  a  succession  (of  enumeral  /3)  of  equal 
and  similarly  ordered  aggregates,  of  which  each  is 
of  enumeral  «,  we  get  a  new  well-ordered  aggregate, 
whose   enumeral  is  defined    to   be  the  product  y8a, 

power — corresponds  to  a  definite  number  of  the  absolutely  infinite 
totality  of  numbers,  and  indeed  reciprocally,  so  that  corresponding  to 
any  transfinite  number  7  there  is  a  (7th)  power  ;  so  that  the  various 
powers  also  form  an  absolutely  infinite  sequence.  This  is  so  much  the 
more  remarkable  as  the  number  7  which  gives  the  rank  of  a  power 
(provided  that  7  has  an  immediate  predecessor)  stands,  to  the  numbers 
of  that  number-class  which  has  this  power,  in  a  magnitude-relation 
whose  smallness  mocks  all  description, — and  this  the  more  7  is  taken  to 
be  greater." 

*  With  this  is  connected  the  promise  to  prove  later  that  the  power  c)f 
the  continuum  is  that  of  (11),  as  stated,  of  course  in  other  words,  in  1878. 
See  the  Notes  at  the  end  oi^  this  book. 


^4  INTRODUCTION 

where  ^  is  the  multiplier  and  a  the  multiplicand. 
Here  also  /3a  is,  in  general,  different  from  a^  ;  but 
we  have,  in  general, 

a(,%)  =  (a/3)y. 

Cantor  also  promised  an  investigation  of  the 
''prime  number-property  "  of  some  of  the  transfinite 
numbers  *  a  proof  of  the  non-existence  of  infinitely 
small  numbers,!  and  a  proof  that  his  previous 
theorem  on  a  point-aggregate  P  in  an  ^2-dimensional 
domain  that,  if  the  derivate  P^'^^  where  «  is  any 
integer  of  (I)  or  (11),  vanishes,  P',  and  hence  P,  is 
of  the  first  power,  can  be  thus  inverted  :  If  P  is 
such  a  point-aggregate  that  P'  is  of  the  first  power, 
there  is  an  integer  a  of  (1)  or  (II)  such  that  P('^>  =  o, 
and  there  is  a  smallest  of  such  a's.  This  last 
theorem  shows  the  importance  of  the  transfinite 
numbers  in  the  theory  of  point-aggregates. 

Cantor's  proof  that  the  power  of  (II)  is  different 
from  that  of  (I)  is  analogous  to  his  proof  of  the 
non-enumerability  of  the  continuum.  Suppose  that 
we  could  put  (II)  in  the  form  of  a  simple  series  : 

(7)  ai,  02,  .  .  .,  a^,  .  .  ., 

we  shall  define  a  number  which  has  the  properties 
both  of  belonging  to  (II)  and  of  not  being  a  member 
of  the  series  (7)  ;  and,  since  these  properties  are 
contradictory  of  one  another  if  the  hypothesis  be 
granted,  we  must  conclude  that  (II)  cannot  be  put 

*  The  property  in  question  is:  A  "prime-number"  a  is  such  that 
the  resolution  a.  =  ^y  is  only  possible  when  /8=  i  or  j8  =  a. 
t  See  the  next  section. 


INTRODUCTION  65 

in  the  form  (7),  and  therefore  has  not  the  power  of 
(I).  Let  a^  be  the  first  number  of  (i)  which  is 
greater  than  ai,   a^  the  first  greater   than  a^ ,    and 

i  2 

so  on  ;  so  that  we  have 


and 

I  <  /Cg  <  /C3  <    .    . 

ai<a^^<aK^<  . 

and 

au<a^^  if  v<kk' 

Now  it  may  happen  that,  from  a  certain  number 
a^   on,  all  following  it  in  the  series  (7)  are  smaller 

than  it  ;  then  it  is  evidently  the  greatest.  If,  on 
the  other  hand,  there  is  no  such  greatest  number, 
imagine  the  series  of  integers  from  I  on  and  smaller 
than  ai,  add  to  it  the  series  of  integers  ^a^  and 
>  a„ ,  then  the  series  of  integers  >  a«  and  <  a^ ,  and 
so  on  ;  we  thus  get  a  definite  part  of  successive 
numbers  of  (I)  and  (II)  which  is  evidently  of  the 
first  power,  and  consequently,  by  the  definition  of 
(II),  there  is  a  least  number  jS  of  (II)  which  is 
greater  than  all  of  these  numbers.      Therefore  /3>  a^ 

and  thus  also  ^  >  a^,  and  also  every  number  /3'  <fi 
is  surpassed  in  magnitude  by  certain  numbers  a^  . 

If  there  is  a  greatest  Uk  =y,  then  the  number  y+  i 
is  a  member  of  (II)  and  not  of  (7)  ;  and  if  there  is 
not  a  greatest,  the  number  /3  is  a  member  of  (II) 
and  not  of  (7). 

Further,  the  power  of  (II)  is  the  7iext  greater  to 
that  of  (I),   so  that    no    other    powers  lie   between 

5 


66  INTRODUCTION 

them,  for  any  aggregate  of  numbers  of  (I)  and  (II) 
is  of  the  power  of  (I)  or  (II).  In  fact,  this  aggregate 
Zj,  when  arranged  in  order  of  magnitude,  is  well- 
ordered,  and  may  be  represented  by 

(a/s),    (/5  =  w,  o)  +  I ,  .  .  .  a,  .  .  . ) 

where  we  always  have  P<Q,  where  Q  is  the  first 
number  of  (III);  and  consequently  (a^)  is  either 
finite  or  of  the  power  of  (I)  or  of  that  of  (II), 
quartuin  non  datur.  From  this  results  the  theorem  : 
If  N  is  any  well-defined  aggregate  of  the  second 
power,  M'  is  a  part  of  M  and  M''  is  a  part  of  M', 
and  we  know  that  M''  is  of  the  same  power  as  M, 
then  M'  is  of  the  same  power  as  M,  and  therefore 
as  M'' ;  and  Cantor  remarked  that  this  theorem  is 
generally  valid,  and  promised  to  return  to  it.* 

Though  the  commutative  law  does  not,  in  general, 
hold  with  the  transfinite  numbers,  the  associative 
law  does,  but  the  distributive  law  is  only  generally 
valid  in  the  form  : 

(a  +  /3)y  =  ay  +  ^y, 

where  a  +  ^,  a,  and  /3  are  multipliers,  ''as  we  im- 
mediately recognize  by  inner  intuition." 

The  subtraction,  division,  prime  numbers,  and 
addition  and  multiplication  of  numbers  which  can 
be  put  in  the  form  of  a  rational  and  integral  function 
of  o)  of  the  transfinite  numbers  were  then  dealt  with 

*  From  the  occurrence  of  this  theorem  on  p.  484  of  the  Math.  Ann., 
xlvi,  1895,  which  we  now  know  (see  the  note  on  p.  204  below)  to  have 
been  a  forestalling  of  the  theorem  that  any  aggregate  can  be  well-ordered, 
we  may  conclude  that  this  latter  theorem  was  used  in  this  instance. 


INTRODUCTION  67 

much  In  the  same  way  as  in  the  memoir  of  1897 
translated  below.  In  the  later  memoir  the  subject 
is|itreated  far  more  completely,  and  was  drawn  up 
with  far  more  attention  to  logical  form  than  was  the 
Grundlagen. 

An  interesting  part  of  the  Grundlagen  is  the 
discussion  of  the  conditions  under  which  we  are  to 
regard  the  introduction  into  mathematics  of  a  new 
conception,  such  as  w,  as  justified.  The  result  of 
this  discussion  was  already  indicated  by  the  way  in 
which  Cantor  defined  his  new  numbers  :  "  We  may 
regard  the  whole  numbers  as  '  actual '  in  so  far  as| 
they,  on  the  ground  of  definitions,  take  a  perfectly 
determined  place  in  our  understanding,  are  clearly 
distinguished  from  all  other  constituents  of  our 
thought,  stand  in  definite  relations  to  them,  and 
thus  modify,  in  a  definite  way,  the  substance  of 
our  mind."  We  may  ascribe  *' actuality  "  to  them 
''  in  so  far  as  they  must  be  held  to  be  an  expression 
or  an  image  {Abbild)  of  processes  and  relations  in 
the  outer  world,  as  distinguished  from  the  intellect." 
Cantor's  position  was,  now,  that  while  there  is  no 
doubt  that  the  first  kind  of  reality  always  implies 
the  second,*  the  proof  of  this  is  often  a  most 
difficult  metaphysical  problem  ;  but,  in  pure  mathe- 
matics, we  need  only  consider  the  first  kind  of 
reality,  and  consequently  "mathematics  is,  in  its 
development,   quite    free,   and    only    subject  to  the 

*  This,  according  to  Cantor,  is  a  consequence  of  "the  unity  of  the 
All,  to  which  we  ourselves  belong,"  and  so,  in  pu7'e  mathematics,  we 
need  only  pay  attention  to  the  reality  of  our  conceptions  in  the  first 
Sense,  as  stated  in  the  text. 


68  INTRODUCTION 

self-evident  condition  that  its  conceptions  are  both 
free  from  contradiction  in  themselves  and  stand 
in  fixed  relations,  arranged  by  definitions,  to 
previously  formed  and  tested  conceptions.  In 
particular,  in  the  introduction  of  new  numbers,  it 
is  only  obligatory  to  give  such  definitions  of  them 
as  will  afford  them  such  a  definiteness,  and,  under 
certain  circumstances,  such  a  relation  to  the  older 
numbers,  as  permits  them  to  be  distinguished  from 
one  another  in  given  cases.  As  soon  as  a  number 
satisfies  all  these  conditions,  it  can  and  must  be 
considered  as  existent  and  real  in  mathematics.  In 
this  I  see  the  grounds  on  which  we  must  regard  the 
rational,  irrational,  and  complex  numbers  as  just  as 
existent  as  the  positive  integers." 

There  is  no  danger  to  be  feared  for  science  from 
this  freedom  in  the  formation  of  numbers,  for,  on 
the  one  hand,  the  conditions  referred  to  under  which 
this  freedom  can  alone  be  exercised  are  such  that 
they  leave  only  a  very  small  opportunity  for  arbi- 
trariness ;  and,  on  the  other  hand,  every  mathe- 
matical conception  has  in  itself  the  necessary 
corrective, — if  it  is  unfruitful  or  inconvenient,  it 
shows  this  very  soon  by  its  unusability,  and  is 
then  abandoned. 

To  support  the  idea  that  conceptions  in  pure 
mathematics  are  free,  and  not  subject  to  any 
metaphysical  control.  Cantor  quoted  the  names 
of,  and  the  branches  of  mathematics  founded  by, 
some  of  the  greatest  mathematicians  of  the  nineteenth 
century,    among    which    an    especially    instructive 


INTRODUCTION  69 

example  in  Kummer's  introduction  of  his  *' ideal" 
numbers  into  the  theory  of  numbers.  But  "  applied  " 
mathematics,  such  as  analytical  mechanics  and 
physics,  is  metaphysical  both  in  its  foundations 
and  in  its  ends.  "  If  it  seeks  to  free  itself  of  this, 
as  was  proposed  lately  by  a  celebrated  physicist,* 
it  degenerates  into  a  'describing  of  nature,' which 
must  lack  both  the  fresh  breeze  of  free  mathematical 
thought  and  the  power  of  explanation  and  grounding 
of  natural  appearances. " 

The  note  of  Cantor's  on  the  process  followed  in 
the  correct  formation  of  conceptions  is  interesting. 
In  his  judgment,  this  process  is  everywhere  the  same  ; 
we  posit  a  thing  without  properties,  which  is  at  first 
nothing  else  than  a  name  or  a  sign  A,  and  give  it 
in  order  different,  even  infinitely  many,  predicates, 
whose  meaning  for  ideas  already  present  is  known, 
and  which  may  not  contradict  one  another.  By 
this  the  relations  of  A  to  the  conception  already 
present,  and  in  particular  to  the  allied  ones,  are 
determined  ;  when  we  have  completed  this,  all  the 
conditions  for  the  awakening  of  the  conception  A, 
which  slumbers  in  us,  are  present,  and  it  enters 
completed  into  "existence"  in  the  first  sense;  to 
prove  its  ' '  existence  "  in  the  second  sense  is  then 
a  matter  of  metaphysics. 

This  seems  to  support  the  process  by  which  Heine, 

*  This  is  evidently  Kirchhoff.  As  is  well  known,  Kirchhoft"  pro- 
posed ( Vorlesungen  iiber  mathemaiische  Physik,  vol.  i,  Mechanik, 
Leipzig,  1874)  this.  Cf.  E.  Mach  in  his  prefaces  to  his  Mechanics 
(3rd  ed.,  Chicago  and  London,  1907  ;  Supplementary  Volume,  Chicago 
and  London,  191 5),  and  Popular  Scientijic  Lecitires,  3rd  ed.,  Chicago 
and  London,  1898,  pp.  236-258. 


70  INTR  OD  UCTION 

in  a  paper  partly  inspired  by  his  discussions  with 
Cantor,  defined  the  real  numbers  as  signs,  to  which 
subsequently  various  properties  were  given.  But 
Cantor  himself,  as  we  shall  see  later,  afterwards 
pointed  out  emphatically  the  mistake  into  which 
Kronecker  and  von  Helmholtz  fell  when  they  started 
in  their  expositions  of  the  number-concept  with  the 
last  and  most  unessential  thing — the  ordinal  words 
or  signs — in  the  scientific  theory  of  number  ;  so  that 
we  must,  I  think,  regard  this  note  of  Cantor's  as 
an  indication  that,  at  this  time  (1882),  he  was  a 
supporter  of  the  formalist  theory  of  number, — or  at 
least  of  rational  and  real  non-integral  numbers. 

In  fact,  Cantor's  notions  as  to  what  is  meant 
by  ''existence"  in  mathematics — notions  which 
are  intimately  connected  with  his  introduction  of 
irrational  and  transfinite  numbers — were  in  substance 
identical  with  those  of  Hankel  (1867)  on  "possible 
or  impossible  numbers."  Hankel  was  a  formalist, 
though  not  a  consistent  one,  and  his  theory  was 
criticized  with  great  acuteness  by  Frege  in  1884. 
But  these  criticisms  mark  the  beginning  of  the 
logical  theory  of  mathematics,  Cantor's  earlier  work 
belonging  to  Xh^  formal  stage,  and  his  later  work  to 
what  may  be  called  the  psychological  stage. 

Finally,  Cantor  gave  a  discussion  and  exact  de- 
termination of  the  meaning  of  the  conception  of 
''continuum."  After  briefly  referring  to  the  dis- 
cussions of  this  concept  due  to  Leucippus,  Demo- 
critus,  Aristotle,  Epicurus,  Lucretius,  and  Thomas 
Aquinas,  and  emphasizing  that  we  cannot  begin,  in 


INTRODUCTION  71 

this  determination,  with  the  conception  of  time  or 
that  of  space,  for  these  conceptions  can  only  be 
clearly  explained  by  means  of  a  continuity-concep- 
tion which  must,  of  course,  be  independent  of  them, 
he  started  from  the  /^-dimensional  plane  aritJinietical 
space  G,„  that  is  to  say,  the  totality  of  systems  of 
values 

in  which  every  x  can  receive  any  real  value  from 
—  00  to  -f  00  independently  of  the  others.  Every 
such  system  is  called  an  ''arithmetical  point"  of 
G„,  the  "distance"  of  two  such  points  is  defined 
by  the  expression 

+  x/{(-^'i-n)'+(-^-'2-^.)'+  •  •  •  +(;^„-^,y), 

and  by  an  ''arithmetical  point-aggregate"  P  con- 
tained in  G„  is  meant  any  aggregate  of  points 
G;,  selected  out  of  it  by  a  law.  Thus  the  investi- 
gation comes  to  the  establishment  of  a  sharp  and 
as  general  as  possible  a  definition  which  should 
allow  us  to  decide  when  P  is  to  be  called  a  ' '  con- 
tinuum." 

If  the  first  derivative  P'  is  of  the  power  of  (I), 
there  is  a  first  number  a  of  (I)  or  (II)  for  which 
p(*)  vanishes  ;  but  if  P'  is  not  of  the  power  of  (I), 
V  can  be  always,  and  in  only  one  way,  divided  into 
two  aggregates  R  and  S,  where  R  is  "reducible," 
— that  is  to  say,  such  that  there  is  a  first  number  y 
of  (I)  or  (II)  such  that 

R(v)  =  o,— 


72  INTRODUCTION 

and  S  is  such  that  derivation  does  not  alter  it.     Then 

and  consequently  also 

and  S  is  said  to  be  "perfect."  No  aggregate  can 
be  both  reducible  and  perfect,  "but,  on  the  other 
hand,  irreducible  is  not  so  much  as  perfect,  nor 
imperfect  exactly  the  same  as  reducible,  as  we 
easily  see  with  some  attention." 

Perfect  aggregates  are  by  no  means  always  every- 
where dense  ;  an  example  of  such  an  aggregate 
which  is  everywhere  dense  in  no  interval  was  given 
by  Cantor.  Thus  such  aggregates  are  not  fitted 
for  the  complete  definition  of  a  continuum,  although 
we  must  grant  that  the  continuum  must  be  perfect. 
The  other  predicate  is  that  the  aggregate  must  be 
connected  (zusaminenhdngend),  that  is  to  say,  if  t 
and  f  are  any  two  of  its  points  and  e  a  given  arbi- 
trarily small  positive  number,  a  finite  number  of 
points  /j,  ^2?  •  •  •)  ^^  of  P  exist  such  that  the  dis- 
tances /^i,    t^t^^    .  .  .,  K^'  3.re  all  less  than  e. 

"All  the  geometric  point-continua  known  to  us 
are,  as  is  easy  to  see,  connected  ;  and  I  believe, 
now,  that  I  recognize  in  these  two  predicates 
'perfect'  and  'connected'  the  necessary  and  sufficient 
characteristics  of  a  point-continuum." 

Bolzano's  (185 1)  definition  of  a  continuum  is 
certainly  not  correct,  for  it  expresses  only  07ie 
property  of  a  continuum,  which  is  also  possessed  by 


INTRODUCTION  73 

aggregates  which  arise  from  G„  when  any  isolated 
aggregate  is  removed  from  it,  and  also  in  those 
consisting  of  many  separated  continua.  Also 
Dedekind  *  appeared  to  Cantor  only  to  emphasize 
rt;;/^///^;- property  of  a  continuum,  namely,  that  which 
it  has  in  common  with  all  other  perfect  aggregates. 

We  will  pass  over  the  development  of  the  theory 
of  point-aggregates  subsequently  to  1882 — Ben- 
dixson's  and  Cantor's  researches  on  the  power  of 
perfect  aggregates.  Cantor's  theory  of  ' '  adherences  " 
and  "coherences,"  the  investigations  of  Cantor, 
Stolz,  Harnack,  Jordan,  Borel,  and  others  on  the 
"content"  of  aggregates,  and  the  applications  of 
the  theory  of  point-aggregates  to  the  theory  of 
functions  made  by  Jordan,  Broden,  Osgood,  Baire, 
Arzela,  Schoenflies,  and  many  others, — and  will  now 
trace  the  development,  in  Cantor's  hands,  of  the 
theory  of  the  transfinite  cardinal  and  ordinal  numbers 
from  1883  to  1895. 

VIII 

An  account  of  the  development  that  the  theory 
of  transfinite  numbers  underwent  in  Cantor's  mind 
from  1883  to  1890  is  described  in  his  articles 
published  in  the  ZeitscJirift  fiir  PJiilosophie  tmd 
pJiilosophiscJie  Kritik  for  1887  and  1888,  and 
collected  and  published  in  1890  under  the  title  Zur 
Lehre  vojh  Transfiniten.  A  great  part  of  this  little 
book  is  taken  up  with  detailed  discussions  about 
philosophers'  denials  of  the  possibility  of  infinite 
*  Essays  on  Number^  p.  U. 


74  INTRODUCTION 

numbers,  extracts  from  letters  to  and  from  philo- 
sophers and  theologians,  and  so  on.*  "All  so- 
called  proofs  of  the  impossibility  of  actually  infinite 
numbers,"  said  Cantor,  "are,  as  may  be  shown  in 
every  particular  case  and  also  on  general  grounds, 
false  in  that  they  begin  by  attributing  to  the 
numbers  in  question  all  the  properties  of  finite 
numbers,  whereas  the  infinite  numbers,  if  they  are 
to  be  thinkable  in  any  form,  must  constitute  quite 
a  new  kind  of  number  as  opposed  to  the  finite 
numbers,  and  the  nature  of  this  new  kind  of  number 
is  dependent  on  the  nature  of  things  and  is  an  object 
of  investigation,  but  not  of  our  arbitrariness  or  our 
prejudice. " 

In  1883  Cantor  had  begun  to  lecture  on  his  view 
of  whole  numbers  and  types  of  order  as  general 
concepts  or  universals  {unuvi  versus  alia)  which 
relate  to  aggregates  and  arise  from  these  aggregates 
when  we  abstract  from  the  nature  of  the  elements. 
"  Every  aggregate  of  distinct  things  can  be  regarded 
as  a  unitary  thing  in  which  the  things  first  mentioned 
are  constitutive  elements.  If  we  abstract  both  from 
the  nature  of  the  elements  and  from  the  order  in 
which  they  are  given,  we  get  the  '  cardinal  number ' 
or  *  power '  of  the  aggregate,  a  general  concept  in 
which  the  elements,  as  so-called  units,  have  so 
grown  organically  into  one  another  to  make  a 
unitary  whole  that  no  one  of  them  ranks  above  the 
others.  Hence  results  that  two  different  aggregates 
have  the  same  cardinal  number  when  and  only  when 
*   Cy.  §  VII,  near  the  beginning. 


INTRODUCTION  75 

they  are  what  I  call  '  equivalent  '  to  one  another, 
and  there  is  no  contradiction  when,  as  often  happens 
with  infinite  aggregates,  two  aggregates  of  which 
one  is  a  part  of  the  other  have  the  same  cardinal 
number.  I  regard  the  non-recognition  of  this  fact 
as  the  principal  obstacle  to  the  introduction  of 
infinite  numbers.  If  the  act  of  abstraction  referred 
to,  when  we  have  to  do  with  an  aggregate  ordered 
according  to  one  or  many  relations  (dimensions),  is 
only  performed  with  respect  to  the  nature  of  the 
elements,  so  that  the  ordinal  rank  in  which  these 
elements  stand  to  one  another  is  kept  in  the  general 
concept,  the  organic  whole  arising  is  what  1  call 
'  ordinal  type,'  or  in  the  special  case  of  well-ordered 
aggregates  an  'ordinal  number.'  This  ordinal 
number  is  the  same  thing  that  I  called,  in  my 
Grundlage7i  of  1883,  the  '  enumeral  (Anzahl)  of  a 
well-ordered  aggregate.'  Two  ordered  aggregates 
have  one  and  the  same  ordinal  type  if  they  stand 
to  one  another  in  the  relation  of  'similarity,' 
which  relation  will  be  exactly  defined.  These  are 
the  roots  from  which  develops  with  logical  necessity 
the  organism  of  transfinite  theory  of  types  and  in 
particular  of  the  transfinite  ordinal  numbers,  and 
which  I  hope  soon  to  publish  in  a  systematic  form." 

The  contents  of  a  lecture  given  in  1883  were  also 
given  in  a  letter  of  1884.  In  it  was  pointed  out 
that  the  cardinal  number  of  an  aggregate  M  is  the 
general  concept  under  which  fall  all  aggregates 
equivalent  to  M,  and  that  : 

"  One   of  the   most    important    problems   of  the 


^6  INTRODUCTION 

theory  of  aggregates,  which  I  believe  I  have  solved 
as  to  its  principal  part  in  my  Grundlagen^  consists 
in  the  question  of  determining  the  various  powers 
of  the  aggregates  in  the  whole  of  nature,  in  so  far 
as  we  can  know  it.  This  end  I  have  reached  by 
the  development  of  the  general  concept  of  enumeral 
of  well-ordered  aggregates,  or,  what  is  the  same 
thing,  of  the  concept  of  ordinal  number."  The 
concept  of  ordinal  number  is  a  special  case  of  the 
concept  of  ordinal  type,  which  relates  to  any  simply 
or  multiply  ordered  aggregate  in  the  same  way  as 
the  ordinal  number  to  a  well-ordered  aggregate. 
The  problem  here  arises  of  determining  the  various 
ordinal  numbers  in  nature. 

When  Cantor  said  that  he  had  solved  the  chief 
part  of  the  problem  of  determining  the  various 
powers  in  nature,  he  meant  that  he  had  almost 
proved  that  the  power  of  the  arithmetical  continuum 
is  the  same  as  the  power  of  the  ordinal  numbers  of 
the  second  class.  In  spite  of  the  fact  that  Cantor 
firmly  believed  this,  possibly  on  account  of  the  fact 
that  all  known  aggregates  in  the  continuum  had 
been  found  to  be  either  of  the  first  power  or  of  the 
power  of  the  continuum,  the  proof  or  disproof  of 
this  theorem  has  not  even  now  been  carried  out, 
and  there  is  some  ground  for  believing  that  it 
cannot  be  carried  out. 

What  Cantor,  in  his  Grundlagen^  had  noted  as  the 
relation  of  two  well-ordered  aggregates  which  have 
the  same  enumeral  was  here  called  the  relation  of 
*' i?imilarity,"  and  in  the   laws  of   multiplication    of 


INTRODUCTION  77 

two  ordinal  numbers  he  departed  from  the  custom 
followed  in  the  Gnindlagen  and  wrote  the  multiplier 
on  the  right  and  the  multiplicand  on  the  left.  The 
importance  of  this  alteration  is  seen  by  the  fact 
that  we  can  write  :  a^  .a^  =  a^^'^  ;  whereas  we  would 
have  to  write,  in  the  notation  of  the   Grundlagen  : 

At  the  end  of  this  letter,  Cantor  remarked  that 
a)  may,  in  a  sense,  be  regarded  as  the  limit  to  which 
the  variable  finite  whole  number  v  tends.  Here  "  w  is 
the  least  transfinite  ordinal  number  which  is  greater 
than  all  finite  numbers  ;  exactly  in  the  same  way 
that  J 2  \s  the  limit  of  certain  variable,  increasing, 
rational  numbers,  with  this  difference  :  the  difference 
between  J  2  and  these  approximating  fractions  be- 
comes as  small  as  we  wish,  whereas  w  —  v  is  always 
equal  to  w.  But  this  difference  in  no  way  alters  the 
fact  that  0)  is  to  be  regarded  as  as  definite  and  com- 
pleted as  J 2,  and  in  no  way  alters  the  fact  that  00 
has  no  more  trace  of  the  numbers  v  which  tend  to  it 
than  J 2  has  of  the  approximating  fractions.  The 
transfinite  numbers  are  in  a  sense  new  h^rationalities, 
and  indeed  in  my  eyes  the  best  method  of  defining 
finite  irrational  numbers  is  the  same  in  principle  as 
my  method  of  introducing  transfinite  numbers.  We 
can  say  that  the  transfinite  numbers  stand  or  fall 
with  finite  irrational  numbers,  in  their  inmost  being 
they  are  alike,  for  both  are  definitely  marked  off 
modifications  of  the  actually  infinite." 

With  this  is  connected  in  principle  an  extract  from 


78  INTRODUCTION 

a  letter  written  in  1886:  ''Finally  I  have  still  to 
explain  to  you  in  what  sense  I  conceive  the  minimum 
of  the  transfinite  as  limit  of  the  increasing  finite. 
For  this  purpose  we  must  consider  that  the  concept 
of  '  limit '  in  the  domain  of  finite  numbers  has  two 
essential  characteristics.  For  example,  the  number 
I  is  the  limit  of  the  numbers  z^=\  —  \\v^  where  v  is 
a  variable,  finite,  whole  number,  which  increases 
above  all  finite  limits.  In  the  first  place  the 
difference  \—z^  is  a  magnitude  which  becomes  in- 
finitely small ;  in  the  second  place  i  is  the  least  of 
all  numbers  which  are  greater  than  all  magnitudes  z^. 
Each  of  these  two  properties  characterizes  the  finite 
number  i  as  limit  of  the  variable  magnitude  z^ . 
Now  if  we  wish  to  extend  the  concept  of  limit  to 
transfinite  limits  as  well,  the  second  of  the  above 
characteristics  is  used  ;  the  first  must  here  be 
allowed  to  drop  because  it  has  a  meaning  only  for 
finite  limits.  Accordingly  1  call  w  the  limit  of  the 
increasing,  finite,  whole  numbers  v ,  because  to  is  the 
least  of  all  numbers  which  are  greater  than  all  the 
finite  numbers.  But  co  —  1/  is  always  equal  to  co,  and 
therefore  we  cannot  say  that  the  increasing  numbers 

V  come  as  near  as  we  wish  to  w  ;  indeed  any  number 

V  however  great  is  quite  as  far  off  from  co  as  the  least 
finite  number.  Here  we  see  especially  clearly  the 
very  important  fact  that  my  least  transfinite  ordinal 
number  (o,  and  consequently  all  greater  ordinal 
numbers,  lie  quite  outside  the  endless  series  i,  2,  3, 
and  so  on.  Thus  w  is  not  a  maximum  of  the  finite 
numbers,  for  there  is  no  such  thing. " 


INTRODUCTION  79 

In  another  letter  written  in  1886,  Cantor  empha- 
sized another  aspect  of  irrational  numbers.  In  all 
of  the  definitions  of  these  numbers  there  is  used, 
as  is  indeed  essential,  a  special  actually  infinite 
aggregate  of  rational  numbers.  In  both  this  and 
another  letter  of  1886,  Cantor  returned  in  great 
detail  to  the  distinction  between  the  "potential" 
and  "actual"  infinite  of  which  he  had  made  a  great 
point  under  other  names  in  his  Gvundlagen.  The 
potential  infinite  is  a  variable  finite,  and  in  order 
that  such  a  variable  may  be  completely  known,  we 
must  be  able  to  determine  the  domain  of  variability, 
and  this  domain  can  only  be,  in  general,  an  actually 
infinite  aggregate  of  values.  Thus  every  potential 
infinite  presupposes  an  actually  infinite,  and  these 
"domains  of  variability"  which  are  studied  in  the 
theory  of  aggregates  are  the  foundations  of  arith- 
metic and  analysis.  Further,  besides  actually  infinite 
aggregates,  we  have  to  consider  in  mathematics 
natural  abstractions  from  these  aggregates,  which 
form  the  material  of  the  theory  of  transfinite 
numbers. 

In  1885,  Cantor  had  developed  to  a  large  extent 
his  theory  of  cardinal  numbers  and  ordinal  types. 

In  the  fairly  long  paper  which  he  wrote  out,  he 
laid  particular  stress  on  the  theory  of  ordinal  types 
and  entered  into  details  which  he  had  not  published 
before  as  to  the  definition  of  ordinal  type  in  general, 
of  which  ordinal  number  is  a  particular  case.  In 
this  paper  also  he  denoted  the  cardinal  number  of 

an    aggregate    M    by   M,   and    the    ordinal   type  of 


8o  INTRODUCTION 

M  by  M  ;  thus  indicating  by  lines  over  the  letter 
that  a  double  or  single  act  of  abstraction  is  to 
be  performed. 

In  the  theory  of  cardinal  numbers,  he  defined  the 
addition  and  multiplication  of  two  cardinal  numbers 
and  proved  the  fundamental  laws  about  them  in 
much  the  same  way  as  he  did  in  the  memoir  of 
1895  which  is  translated  below.  It  is  characteristic 
of  Cantor's  views  that  he  distinguished  very  sharply 
between  an  aggregate  and  a  cardinal  number  that 
belongs  to  it  :  "Is  not  an  aggregate  an  object  out- 
side us,  whereas  its  cardinal  number  is  an  abstract 
picture  of  it  in  our  mind  ?  " 

In  an  ordered  aggregate  of  any  number  of 
dimensions,  such  as  the  totality  of  points  in  space, 
as  determined  by  three  rectangular  co-ordinates,  or 
a  piece  of  music  whose  dimensions  are  the  sequence 
of  the  tones  in  time,  the  duration  of  each  tone  in 
time,  the  pitch  of  the  tones,  and  the  intensity  of  the 
tones,  then  * '  if  we  make  abstraction  of  the  nature 
of  the  elements,  while  we  retain  their  rank  in  all  the 
n  different  directions,  an  intellectual  picture,  a  general 
concept,  is  generated  in  us,  and  I  call  this  the  /^-ple 
ordinal  type."  The  definition  of  the  "  similarity  of 
ordered  aggregates  "  is  : 

"Two  ;/-ply  ordered  aggregates  M  and  N  are 
called  similar  if  it  is  possible  so  to  make  their 
elements  correspond  to  another  uniquely  and  com- 
pletely that,  if  E  and  E'  are  any  two  elements  of 
M  and  F  and  F'  the  two  corresponding  elements  of 
N,  then  for  i/=  i,  2,  .   .   .  n  the  relation  of  rank  of 


INTRODUCTION  8i 

E  to  E'  in  the  jyth  direction  inside  the  aggregate  M 
is  exactly  the  same  as  the  relation  of  rank  of  F  to  F' 
in  the  v\\\  direction  inside  the  aggregate  N.  We 
will  call  such  a  correspondence  of  two  aggregates 
which  are  similar  to  one  another  an  imaging  of  the 
one  on  the  other." 

The  addition  and  multiplication  of  ordinal  types, 
and  the  fundamental  laws  about  them,  were  then 
dealt  with  much  as  in  the  memoir  of  1895  which  is 
translated  below.  The  rest  of  the  paper  was  devoted 
to  a  consideration  of  problems  about  ;^-ple  finite 
types. 

In  1888,  Cantor,  who  had  arrived  at  a  very  clear 
notion  that  the  essential  part  of  the  concept  of  number 
lay  in  the  unitary  concept  that  we  form,  gave  some 
interesting  criticisms  on  the  essays  of  Helmholtz  and 
Kronecker,  which  appeared  in  1887,  on  the  concept 
of  number.  Both  the  authors  referred  to  started 
with  the  last  and  most  unessential  feature  in  our 
treatment  of  ordinal  numbers  :  the  words  or  other 
signs  that  we  use  to  represent  these  numbers. 

In  1887,  Cantor  gave  a  more  detailed  proof  of  the 
non-existence  of  actually  infinitely  small  magnitudes. 
This  proof  was  referred  to  in  advance  in  the  Grund- 
lagen,  and  was  later  put  into  a  more  rigorous  form 
by  Peano. 

We  have  already  referred  to  the  researches  of 
Cantor  on  point-aggregates  published  in  1883  and 
later  ;  the  only  other  paper  besides  those  already 
dealt  with  that  was  published  by  Cantor  on  an 
important    question    in    the    theory    of    transfinite 


82  •    INTRODUCTION 

numbers  was  one  Ipublished  in  1892.  In  this  paper 
we  can  see  the  origins  of  the  conception  of  '  *  cover- 
ing" {Belegmig)  defined  in  the  memoir  of  1895  trans- 
lated below.  In  the  terminology  introduced  in  this 
memoir,  we  can  say  that  the  paper  of  1892  contains 
a  proof  that  2,  when  exponentiated  by  a  transfinite 
cardinal  number,  gives  rise  to  a  cardinal  number 
which  is  greater  than  the  cardinal  number  first 
mentioned. 

The  introduction  of  the  concept  of  "covering"  is 
the  most  striking  advance  in  the  principles  of  the 
theory  of  transfinite  numbers  from  1885  to  1895, 
and  we  can  now  study  the  final  and  considered  form 
which  Cantor  gave  to  the  theory  in  two  important 
memoirs  of  1895  ^"^  1897.  The  principal  advances 
in  the  theory  since  1897  will  be  referred  to  in  the 
notes  at  the  end  of  this  book. 


CONTRIBUTIONS   TO  THE 
FOUNDING   OF   THE  THEORY  OF 

TRANSFINITE   NUMBERS 


[48i]      CONTRIBUTIONS  TO  THE 
FOUNDING  OF  THE  THEORY  OF 
TRANSFINITE  NUMBERS 

(First  Article) 

"  Hypotheses  non  fingo." 

"  Neque  enim  leges  intellectui  aut  rebus  damus 
ad  arl)itrium  nostrum,  sed  tanquam  scribal 
fideles  ab  ipsius  naturae  voce  latas  et  prolatas 
excipimus  et  describimus." 

"Veniet  tempus,  quo  ista  quae  nunc  latent,  in 
lucem  dies  extrahat  et  longioris  xvi  diligentia." 

The  Conception  of  Power  or  Cardinal  Number 

By  an  ^ '  aggregate  "  {Menge)  we  are  to  understand 
any  collection  into  a  whole  {Zusaimnenfassung  zu 
einem  Ganzen)  M  of  definite  and  separate  objects  ;;/ 
of  our  intuition  or  our  thought.  These  objects  are 
called  the  ' '  elements  "  of  M. 
In  signs  we  express  this  thus  : 

(i)  M  =  {;;/}. 

We  denote  the  uniting  of  many  aggregates  M,  N, 
P,  .  .  .,  which  have  no  common  elements,  into  a 
single  aggregate  by 

(2)  (M,  N,  P,  .  .  .). 

85 


m       THE  FOUNDING  OF  THE   THEORY 

The  elements  of  this  aggregate  are,  therefore,  the 
elements  of  M,  of  N,  of  P,  .  .  .,  taken  together. 

We  will  call  by  the  name  "part"  or  "partial 
3-ggi'egate  "  of  an  aggregate  M  any  other  aggregate 
Mj  whose  elements  are  also  elements  of  M. 

If  M2  is  a  part  of  M^  and  M^  is  a  part  of  M,  then 
M2  is  a  part  of  M. 

Every  aggregate  M  has  a  definite  "  power,"  which 
we  will  also  call  its  "cardinal  number." 

We  will  call  by  the  name  "  power"  or  "cardinal 
number  "  of  M  the  general  concept  which,  by  means 
of  our  active  faculty  of  thought,  arises  from  the 
aggregate  M  when  we  make  abstraction  of  the 
nature  of  its  various  elements  in  and  of  the  order 
in  which  they  are  given. 

[482]  We  denote  the  result  of  this  double  act  of 
abstraction,  the  cardinal  number  or  power  of  M,  by 

(3)  M. 

Since  every  single  element  vi,  if  we  abstract  from 
its  nature,  becomes  a  "unit,"  the  cardinal  number 
M  is  a  definite  aggregate  composed  of  units,  and 
this  number  has  existence  in  our  mind  as  an  intel- 
lectual image  or  projection  of  the  given  aggregate  M. 

We  say  that  two  aggregates  M  and  N  are  ' '  equi- 
valent," in  signs 

(4)  M  00  N     or     N  fV)  M, 

if  it  is  possible  to  put  them,  by  some  law,  in  such  a 
relation  to  one  another  that  to  every  element  of  each 
onelof  them  corresponds  one  and  only  one  element 


OF  TRANSFINITE  NUMBERS  87 

of  the  other.  To  every  part  Mj  of  M  there  corre- 
sponds, then,  a  definite  equivalent  part  N^  of  N,  and 
inversely. 

If  we  have  such  a  law  of  co-ordination  of  two 
equivalent  aggregates,  then,  apart  from  the  case 
when  each  of  them  consists  only  of  one  element,  we 
can  modify  this  law  in  many  ways.  We  can,  for 
instance,  always  take  care  that  to  a  special  element 
in^  of  M  a  special  element  n^  of  N  corresponds.  For 
if,  according  to  the  original  law,  the  elements  ni^ 
and  Uq  do  not  correspond  to  one  another,  but  to  the 
element  m^  of  M  the  element  n^^  of  N  corresponds, 
and  to  the  element  n^  of  N  the  element  in^  of  M 
corresponds,  we  take  the  modified  law  according  to 
which  m^  corresponds  to  n^  and  m^  to  n^  and  for  the 
other  elements  the  original  law  remains  unaltered. 
By  this  means  the  end  is  attained. 

Every  aggregate  is  equivalent  to  itself : 

(5)  M  00  M. 

If  two  aggregates  are  equivalent  to  a  third,  they  are 
equivalent  to  one  another  ;  that  is  to  say  : 

(6)  from  M  00  P    and    N  00  P    follows    M  00  N. 

Of  fundamental  importance  is  the  theorem  that 
two  aggregates  M  and  N  have  the  same  cardinal 
number  if,  and  only  if,  they  are  equivalent  :  thus, 

(7)  from     M  00  N     we  get     M  =  N, 
and 

(8)  from     M  =  N     we  get     M  00  N. 

Thus  the  equivalence  of  aggregates  forms  the  neces- 


88       THE  FOUNDING  OF  THE  THEORY 

sary  and  sufficient  condition  for  the  equality  of  their 
cardinal  numbers. 

[483]  In  fact,  according  to  the  above  definition  of 
power,  the  cardinal  number  M  remains  unaltered  if 
in  the  place  of  each  of  one  or  many  or  even  all 
elements  in  of  M  other  things  are  substituted.  If, 
now,  M  00  N,  there  is  a  law  of  co-ordination  by 
means  of  which  M  and  N  are  uniquely  and  recipro- 
cally referred  to  one  another ;  and  by  it  to  the 
element  in  of  M  corresponds  the  element  n  of  N. 
Then  we  can  imagine,  in  the  place  of  every  element 
m  of  M,  the  corresponding  element  /^  of  N  substi- 
tuted, and,  in  this  way,  M  transforms  into  N  without 
alteration  of  cardinal  number.      Consequently 

M  =  N. 

The  converse  of  the  theorem  results  from  the  re- 
mark that  between  the  elements  of  J^I  and  the 
different  units  of  its  cardinal  number  M  a  recipro- 
cally univocal  (or  bi-univocal)  relation  of  correspond- 
ence subsists.  For,  as  we  saw,  M  grows,  so  to 
speak,  out  of  M  in  such  a  way  that  from  every 
element  ;;/  of  M  a  special  unit  of  M  arises.  Thus 
we  can  say  that 

(9)  M  00  M. 

In  the  same  way  N  00  N.  If  then  M  =  N,  we  have, 
by  (6),  M  00  N. 

We  will  mention  the   following   theorem,   which 
results  immediately  from  the  conception  of  equival- 


OF  TRANSFINITE  NUMBERS  89 

ence.  If  M,  N,  P,  .  .  .  are  aggregates  which  have 
no  common  elements,  M',  N",  P',  .  .  .  are  also  aggre- 
gates with  the  same  property,  and  if 

M  00  M',     N  00  N;     P  cv)  P^     .  .  . , 

then  we  always  have 

(M,  N,  P,  .  .  .)  fNJ  (M;  N',  P;  .  .  .). 

§2 

*' Greater"  and  <*  Less  "  with  Powers 

If  for  two  aggregates  M  and  N  with  the  cardinal 
numbers  a=M  and  b  =  N,  both  the  conditions  : 

{a)  There  is  no  part  of  M  which  is  equivalent  to  N, 
{b)  There  is  a  part  N^  of  N,  suCh  that  N^  00  M, 

are  fulfilled,  it  is  obvious  that  these  conditions  still 
hold  if  in  them  M  and  N  are  replaced  by  two 
equivalent  aggregates  M'  and  N'.  Thus  they  ex- 
press a  definite  relation  of  the  cardinal  numbers 
a  and   b  to  one  another. 

[484]  P'urther,  the  equivalence  of  M  and  N,  and 
thus  the  equality  of  a  and  b,  is  excluded  ;  for  if  we 
had  M  00  N,  we  would  have,  because  N^  f\j  M,  the 
equivalence  N^  po  N,  and  then,  because  M  00  N, 
there  would  exist  a  part  Mj  of  M  such  that  M^  oo  M, 
and  therefore  wg  should  have  Mj  oo  N  ;  and  this 
contradicts  the  condition  {a). 

Thirdly,  the  relation  of  a  to  b  is  such  that  it 
makes  impossible  the  same  relation  of  b  to  vi ;  for  if 


90       THE  FOUNDING  OF  THE   THEORY 

in  ia)  and  {b)  the  parts  played  by  M  and  N  are 
interchanged,  two  conditions  arise  which  are  con- 
tradictory to  the  former  ones. 

We  express  the  relation  of  a  to  b  characterized  by 
{a)  and  ip)  by  saying:  a  is  "less"  than  b  or  b  is 
"greater  "  than  a  ;  in  signs 

(i)  a<b     or     \i>(x. 

We  can  easily  prove  that, 

(2)  if  a<b  and  b<c,  then  we  always  have  a<c. 
Similarly,    from   the   definition,    it   follows   at  once 
that,  if  ?i  is  part  of  an   aggregate   P,  from  a<Pi 

follows  a  <  P  and  from  P  <  b  follows  P^  <  b. 
We  have  seen  that,  of  the  three  relations 

a  =  b,  (x<^,  \i<(x, 

each  one  excludes  the  two  others.  On  the  other 
hand,  the  theorem  that,  with  any  two  cardinal 
numbers  a  and  b,  one  of  those  three  relations  must 
necessarily  be  realized,  is  by  no  means  self-evident 
and  can  hardly  be  proved  at  this  stage. 

Not  until  later,  when  we  shall  have  gained  a 
survey  over  the  ascending  sequence  of  the  transfinite 
cardinal  numbers  and  an  insight  into  their  connexion, 
will  result  the  truth  of  the  theorem  : 

A.  If  a  and  b  are  any  two  cardinal  numbers,  then 
either  a  =  b  or  a  <  b  or  a  >  b. 

P>om  this  theorem  the  following  theorems,  of 
which,  however,  we  will  here  make  no  use,  can  be 
very  simply  derived  : 


OF  TRANSFINITE  NUMBERS  91 

B.  If  two  aggregates  M  and  N  are  such  that  M  is 
equivalent  to  a  part  N^  of  N  and  N  to  a  part  Mj  of 
M,  then  M  and  N  are  equivalent  ; 

C.  If  Mj  is  a  part  of  an  aggregate  M,  Mg  is  a 
part  of  the  aggregate  Mj,  and  if  the  aggregates 
M  and  Mg  are  equivalent,  then  Mj  is  equivalent  to 
both  M  and  Mg  ; 

D.  If,  with  two  aggregates  M  and  N,  N  is 
equivalent  neither  to  M  nor  to  a  part  of  M,  there  is 
a  part  Nj  of  N  that  is  equivalent  to  M  ; 

E.  If  two  aggregates  M  and  N  are  not  equivalent, 
and  there  is  a  part  N^  of  N  that  is  equivalent  to  M, 
then  no  part  of  M  is  equivalent  to  N. 

[485]  §  3 

The  Addition  and  Multiplication  of  Powers 

The  union  of  two  aggregates  M  and  N  which 
have  no  common  elemeijts  was  denoted  in  §  i,  (2), 
by  (M,  N).  We  call  it  the  ''union-aggregate 
( Vereinigungsinenge)  of  M  and  N. " 

If  M'  and  N'  are  two  other  aggregates  without 
common  elements,  and  if  M  00  M'  and  N  00  N',  we 
saw  that  we  have 

(M,  N)  00  (M',  N'). 
Hence  the  cardinal  number  of  (M,  N)  only  depends 

upon  the  cardinal  numbers  M  =  a  and  N  =  b. 

This  leads  to  the  definition  of  the  sum  of  vi  and  b. 
We  put  ' 

(I)  a  +  b  =  (M7>^). 


92       THE  FOUNDING  OF  THE   THEORY 

Since  in  the  conception  of  power,  we  abstract  from 
the  order  of  the  elements,  we  conclude  at  once  that 

(2)  a  +  b  =  b  +  a; 

and,  for  any  three  cardinal  numbers  a,  b,  c,  we  have 

(3)  a  +  (b  +  c)  =  (a  +  b)  +  c. 

We  now  come  to  multiplication.  Any  element  vi 
of  an  aggregate  M  can  be  thought  to  be  bound  up 
with  any  element  n  of  another  aggregate  N  so  as 
to  form  a  new  element  {m,  n)  ;  we  denote  by  (M  .  N) 
the  aggregate  of  all  these  bindings  (w,  n)^  and  call 
it  the  ''aggregate  of  bindings  {Verbindungsniejige) 
ofMandN."     Thus 

(4)  (M.N)  =  {K;.)}. 

We  see  that  the  power  of  (M  .  N)  only  depends  on 

the  powers  M  =  a  and  N  =  b  ;  for,  if  we  replace  the 
aggregates  M  and  N  by  the  aggregates 

W  =  {m)     and     N'  =  {;/} 

respectively  equivalent  to  them,  and  consider  in,  in' 
and  ft,  n'  as  corresponding  elements,  then  the 
aggregate 

(M'.N')  =  {(^^/,  n)) 

is  brought  into  a  reciprocal  and  univocal  corre- 
spondence with  (M .  N)  by  regarding  {in,  n)  and 
{in' ,   n')  as  corresponding  elements.      Thus 

(5)  (m;no<^^(m.n). 

We  now  define  the  product  vi .  b  by  the  equation 


(6)  a.b  =  (M.N). 


OF  TRANSFINITE  NUMBERS  93 

[486]  An  aggregate  with  the  cardinal  number 
a .  b  may  also  be  made  up  out  of  two  aggregates  M 
and  N  with  the  cardinal  numbers  a  and  b  according 
to  the  following  rule  :  We  start  from  the  aggregate 
N  and  replace  in  it  every  element  n  by  an  aggregate 
M„  fNJ  M  ;  if,  then,  we  collect  the  elements  of  all 
these  aggregates  M„  to  a  whole  S,  we  see  that 

(7)  S  ro  (M  .  N), 
and  consequently 

S  =  a.b. 

For,  if,  with  any  given  law  of  correspondence  of  the 
two  equivalent  aggregates  M  and  M,,,  we  denote 
by  in  the  element  of  M  which  corresponds  to  the 
element  m^  of  M„,  we  have 

(8)  S^  {;;/.}; 

and  thus  the  aggregates  S  and  (M  .  N)  can  be  re- 
ferred reciprocally  and  univocally  to  one  another  by 
regarding  ni^  and  {in,  n)  as  corresponding  elements. 
From  our  definitions  result  readily  the  theorems  : 

(9)  a.b  =  b.a, 

(10)  a.(b  .  c)  =  (a.  b).  c, 

(11)  a(b  +  c)  =  ab  +  ac; 
because  : 

(M.N)r>o(N.M), 

(M.(N.P))  00  ((M.N).  P), 
(M  .  (N,  P))  fNj  ((M  .  N),  (M  .  P)). 
Addition  and  multiplication  of  powers  arc  subject, 


94       THE  FOUNDING  OF  THE  THEORY 

therefore,  to  the  commutative,  associative,  and  dis- 
tributive laws. 


§4 
The  Exponentiation  of  Powers 

By  a  "  covering  of  the  aggregate  N  with  elements 
of  the  aggregate  M,"  or,  more  simply,  by  a  ''cover- 
ing of  N  with  M,"  we  understand  a  law  by  which 
with  every  element  n  of  N  a  definite  element  of  M 
is  bound  up,  where  one  and  the  same  element  of  M 
can  come  repeatedly  into  application.  The  element 
of  M  bound  up  with  n  is,  in  a  way,  a  one-valued 
function  of  n,  and  may  be  denoted  by  f{}i)  ;  it  is 
called  a  ''  covering  function  of  n.''  The  correspond- 
ing covering  of  N  will  be  called /(N). 

[487]  Two  coverings /i(N)  and/2(N)  are  said  to 
be  equal  if,  and  only  if,  for  all  elements  ;^  of  N  the 
equation 

(I)  /lW=/2(«) 

is  fulfilled,  so  that  if  this  equation  does  not  subsist 
for  even  a  single  element  n^n^,  f^{^)  and/2(N)  are 
characterized  as  different  coverings  of  N.  For  ex- 
ample, if  vi^  is  a  particular  element  of  M,  we  may 
fix  that,  for  all  n'?> 

f{n)  =  m^; 

this  law  constitutes  a  particular  covering  of  N  with 
M.  Another  kind  of  covering  results  if  ui^  and  m^ 
are  two  different  particular  elements  of  M  and  n^  a 
particular  element  of  N,  from  fixing  that 


OF  TRANSFINITE  NUMBERS  95 

/(«o)  =  ^'^o 

for  all  ;2's  which  are  different  from  n^. 

The  totality  of  different  coverings  of  N  with  M 
forms  a  definite  aggregate  with  the  elements /(N)  ; 
we  call  it  the  **  covering-aggregate  (yBelegungsvienge) 
of  N  with  M  "  and  denote  it  by  (N  |  M).       Thus  : 

(2)  (N  |M)={/(N)}. 

If  M  00  M'  and  N  Oo  N',  we  easily  find  that 

(3)  (N  I  M)  00  (N'  I  MO. 

Thus  the  cardinal  number  of  (N  |  M)  depends  only 
on  the  cardinal  numbers  M  =  a  and  N  =  b  ;  it  serves 
us  for  the  definition  of  a*  : 

(4)  a»  =  (N^r). 

For  any  three  aggregates,  M,  N,  P,  we  easily  prove 
the  theorems: 

.     (5)      ((N  |M).(P|M))f\j((N,  P)|M), 

(6)  ((P|  M).(P|N))<N;(P|(M.N)), 

(7)  (P|(N|  M))'\^((P.N)|M), 

from  which,  if  we  put  P  =  c,  we  have,  by  (4)  and  by 
paying  attention  to  §  3,  the  theorems  for  any  three 
cardinal  numbers,  a,  b,  and  c : 


(8) 

at.rt'  =  a''+^ 

(9) 

a'.b^  =  (a.b)f, 

(10) 

(a*/ = '!'•'• 

96       THE  POUNDING  OF  THE   THEORY 

[488]  We  see  how  pregnant  and  far-reaching 
these  simple  formulae  extended  to  powers  are  by  the 
following  example.  If  we  denote  the  power  of  the 
linear  continuum  X  (that  is,  the  totality  X  of  real 
numbers  x  such  that  x>^  and  <i)  by  0,  we  easily 
see  that  it  may  be  represented  by,  amongst  others, 
the  formula  : 

(II)  r     o  =  2^\ 

where  §  6  gives  the  meaning  of  Nq.  In  fact,  by  (4), 
2^0  is  the  power  of  all  representations 

(-)     .=^+f>+...+f>+... 

(where  f{v)  =  O  or  i ) 

of  the  numbers  x  in  the  binary  system.  If  we  pay 
attention  to  the  fact  that  every  number  x  is  only 
represented  once,  with  the  exception  of  the  numbers 

x= <i,  which  are  represented  twice  over,  we 

have,  if  we  denote  the  "enumerable"  totality  of 
the  latter  by  {s^], 


2«o  =  ({j4,    X). 

If  we  take  away  from  X  any  "  enumerable  "  aggre- 
gate {t^}  and  denote  the  remainder  by  X^,  we  have  : 

({sA,  X)=({^4,  {4},  X,), 

so 

Xco(K},  X), 


OF  TRANSFINlTli  NUMBERS  97 

and  thus  (§  i) 

2^"  =  X  =  0. 

From  (11)  follows  by  squaring  (by  §  6,  (6)) 

and  hence,  by  continued  multiplication  by  0, 

(13)  o''=o, 

where  v  is  any  finite  cardinal  number. 

If  we  raise  both  sides  of  (11)  to  the  power*  ^j^ 
we  get  ^ 

But  since,  by  §  6,  (8),  No.^^o  =  ^^o,  we  have 

(14)  o^''*°  =  o. 

The  formulae  (13)  and  (14)  mean  that  both  the 
jy-dimensional  and  the  {s?Q-dimensional  continuum  have 
the  power  of  the  one-dimensional  continuum.  Thus 
the  whole  contents  of  my  paper  in  Crelle's  Journal^ 
vol.  Ixxxiv,  i878,t  are  derived  purely  algebraically 
with  these  few  strokes  of  the  pen  from  the  fundamental 
formulae  of  the  calculation  with  cardinal  numbers. 

[489]  §  5 

The  Finite  Cardinal  Numbers 

We  will  next  show  how  the  principles  which  we 
have  laid  down,  and  on  which  later  on  the  theory 
of  the  actually  infinite  or  transfinite  cardinal  numbers 

*  [In  English  there  is  an  ambiguity.] 
t  [See  Section  V  of  the  Introduction.] 


98       THE  FOUNDING  OF  THE  THEORY 

will  be  built,  afford  also  the  most  natural,  shortest, 
and  most  rigorous  foundation  for  the  theory  of 
finite  numbers. 

To  a  single  thing  e^,  if  we  subsume  it  under  the 
concept  of  an  aggregate  Eo  =  (^o)'  corresponds  as 
cardinal  number  what  we  call  ''one"  and  denote  by 
I  ;  we  have 

(1)  i  =  Eo. 

Let  us  now  unite  with  E^  another  thing  e^,  and 
call  the  union-aggregate  E^,  so  that 

(2)  t:i  =  (Ko,  e^  =  {e^,  e^). 

The  cardinal  number  of  E^  is  called  "two"  and  is 
denoted  by  2  : 

(3)  2=1,. 

By  addition  of  new  elements  we  get  the  series  of 
aggregates 

E2  =  (Ei,  ^2)'      E3  =  (E2,  ^3),  .  .  ., 

which  give  us  successively,  in  unlimited  sequence, 
the  other  so-called  ' '  finite  cardinal  numbers  "  de- 
noted by  3,  4,  5,  .  .  .  The  use  which  we  here 
make  of  these  numbers  as  suffixes  is  justified  by 
the  fact  that  a  number  is  only  used  as  a  suffix 
when  it  has  b"een  defined  as  a  cardinal  number. 
We  have,  if  by  i/— i  is  understood  the  number  im- 
mediately preceding  v  in  the  above  series, 

(4)  j/  =  E,_i, 

(5)  E,-(E,_i,  O  =  (^o'  ^D  •  •  •  ^^)' 


OF  TRANSFINITE  NUMBERS  99 

From  the  definition  of  a  sum  in  §  3  follows  : 

(6)  E,-E,_i+i; 

that  is  to  say,  every  cardinal  number,  except  i,  is 
the  sum  of  the  immediately  preceding  one  and  i. 

Now,  the  following  three  theorems  come  into  the 
foreground  : 

A.  The  terms  of  the  unlimited  series  of  finite 
cardinal  numbers 


are  all  different  from  one  another  (that  is  to  say, 
the  condition  of  equivalence  established  in  §  I  is 
not  fulfilled  for  the  corresponding  aggregates). 

[490]  B.  Every  one  of  these  numbers  v  is  greater 
than  the  preceding  ones  and  less  than  the  following 
ones  (§2). 

C.  There  are  no  cardinal  numbers  which,  in 
magnitude,  lie  between  two  consecutive  numbers 
V  and  J^+  I  (§  2). 

We  make  the  proofs  of  these  theorems  rest  on 
the  two  following  ones,  D  and  E.  We  shall,  then, 
in  the  next  place,  give  the  latter  theorems  rigid 
proofs. 

D.  If  M  is  an  aggregate  such  that  it  is  of  equal 
power  with  none  of  its  parts,  then  the  aggregate 
(M,  e),  which  arises  from  M  by  the  addition  of  a 
single  new  element  e^  has  the  same  property  of 
being  of  equal  power  with  none  of  its  parts. 

E.  If  N  is  an  aggregate  with  the  finite  cardinal 
number  ^z,    and   N^  is  any  part    of  N,    the  cardinal 


lOO     THE  FOUNDING  OF  THE   THEORY 

number  of  Nj,  is  equal  to  one  of  the  preceding 
numbers  i,  2,  3,  .  .  . ,  ly—  i. 

Pr^^/^/D.— Suppose  that  the  aggregate  (M,  e) 
is  equivalent  to  one  of  its  parts  which  we  will  call 
N.  Then  two  cases,  both  of  which  lead  to  a  con- 
tradiction, are  to  be  distinguished  : 

{(i)  The  aggregate  N  contains  e  as  element  ;  let 
N  =  (Ml,  e)  ;  then  M^  is  a  part  of  M  because  N  is 
a  part  of  (M,  e).  As  we  saw  in  §  i,  the  law  of 
correspondence  of  the  two  equivalent  aggregates 
(M,  e)  and  (M^,  e)  can  be  so  modified  that  the 
element  e  of  the  one  corresponds  to  the  same 
element  e  of  the  other  ;  by  that,  then,  M  and  M^ 
are  referred  reciprocally  and  univocally  to  one 
another.  But  this  contradicts  the  supposition  that 
M  is  not  equivalent  to  its  part  M^. 

{U)  The  part  N  of  (M,  e)  does  not  contain  e  as 
element,  so  that  N  is  either  M  or  a  part  of  M.  In 
the  law  of  correspondence  between  (M,  e)  and  N, 
which  lies  at  the  basis  of  our  supposition,  to  the 
element  e  of  the  former  let  the  element  /  of  the 
latter  correspond.  Let  N  =  (Mi,  /)  ;  then  the 
aggregate  M  is  put  in  a  reciprocally  univocal  relation 
with  Mj.  But  Ml  is  a  part  of  N  and  hence  of  M. 
So  here  too  M  would  be  equivalent  to  one  of  its 
parts,  and  this  is  contrary  to  the  supposition. 

Proof  of  E. — We  will  suppose  the  correctness 
of  the  theorem  up  to  a  certain  v  and  then  conclude 
its  validity  for  the  number  v\-  i  which  immediately 
follows,  in  the  following  manner  : — We  start  from 
the  aggregate  E„  =  (^o.  ^1,  •  •  -,  ^.)  as  an   aggregate 


OF  TRANSFINITE  Ntl'MBERTy        (ot 

with  the  cardinal  number  v-\-\.  If  the  theorem  is 
true  for  this  aggregate,  its  truth  for  any  other 
aggregate  with  the  same  cardinal  number  v-\-\ 
follows  at  once  by  §  i.  Let  E'  be  any  part  of  E^  ; 
we  distinguish  the  following  cases  : 

{a)  E'  does  not  contain  e^  as  element,  then  E  is 
either  E^_i  [491]  or  a  part  of  E^,_i,  and  so  has  as 
cardinal  number  either  v  or  one  of  the  numbers 
I,  2,  3,  .  .  .,  I/—  I,  because  we  supposed  our  theorem 
true  for  the  aggregate  E^_i,  with  the  cardinal 
number  v. 

{b)  E'  consists   of   the    single    element    ^„,    then 

E'=i. 

ic)  E'  consists  of  e^  and  an  aggregate  E'',  so  that 
E'  =  (E'',  e^,  E''  is  a  part  of  E^_i  and  has  there- 
fore by  supposition  as  cardinal  number  one  of  the 
numbers  i,  2,  3,  .  .  .,  1/— i.  But  now  E'  =  E''4-i, 
and  thus  the  cardinal  number  of  E'  is  one  of  the 
numbers  2,  3,  .  .  . ,  i/. 

Proof  of  K. — Every  one  of  the  aggregates  which 
we  have  denoted  by  E^  has  the  property  of  not 
being  equivalent  to  any  of  its  parts.  Eor  if  we 
suppose  that  this  is  so  as  far  as  a  certain  i/,  it  follows 
from  the  theorem  D  that  it  is  so  for  the  immediately 
following  number  v-\-\.  For  1/=  i,  we  recognize  at 
once  that  the  aggregate  Ei  =  (^o,  e^  is  not  equivalent 
to  any  of  its  parts,  which  are  here  {e^  and  {e^. 
Consider,  now,  any  two  numbers  fx  and  v  of  the 
series  1,2,  3,  .  .  .  ;  then,  if  jx  is  the  earlier  and  v 
the  later,   E^_i  is  a  part  of  E^_i.      Thus  E^_i  and 


i02TliE  FOUNDING  OF  THE   THEORY 

E^_i    are     not    equivalent,    and     accordingly    their 

cardinal  numbers  ^=:E^_i  and  i/=E^_i  are  not 
equal. 

Proof  of  ^. — If  of  the  two  finite  cardinal  numbers 
fj.  and  V  the  first  is  the  earlier  and  the  second  the 
later,  then  fj.  <  v.  For  consider  the  two  aggregates 
M  =  E^_i  and  N  =  E^,_i;  for  them  each  of  the  two 

conditions  in  §  2  for  M  <  N  is  fulfilled.  The  con- 
dition {a)  is  fulfilled  because,  by  theorem  E,  a  part 
of  M  =  E^_i  can  only  have  one  of  the  cardinal 
numbers  i,  2,  3,  .  .  .,  /x— i,  and  therefore,  by 
theorem  A,  cannot  be  equivalent  to  the  aggregate 
N=E^_i.  The  condition  (b)  is  fulfilled  because  M 
itself  is  a  part  of  N. 

Proof  of  C. — Let  a  be  a  cardinal  number  which 
is  less  than  v-\-  i.  Because  of  the  condition  (b)  of 
§  2,  there  is  a  part  of  E,,  with  the  cardinal  number 
a.  By  theorem  E,  a  part  of  E^,  can  only  have  one 
of  the  cardinal  numbers  i,  2,  3,  .  .  . ,  i/.  Thus  a  is 
equal  to  one  of  the  cardinal  numbers  I,  2,  3,  .  .  .,  i/. 
By  theorem  B,  none  of  these  is  greater  than  v. 
Consequently  there  is  no  cardinal  number  a  which 
is  less  than  v+  i  and  greater  than  v. 

Of  importance  for  what  follows,  is  the  following 
theorem  : 

F.  If  K  is  any  aggregate  of  different  finite 
cardinal  numbers,  there  is  one,  /c^,  amongst  them 
which  is  smaller  than  the  rest,  and  therefore  the 
smallest  of  all. 

[492]   Proof — The  aggregate   K  either  contains 


OF  TRANSFINITE  NUMBERS        103 

the  number  i,  in  which  case  it  is  the  least,  «:i=i, 
or  it  does  not.  In  the  latter  case,  let  J  be  the 
aggregate  of  all  those  cardinal  numbers  of  our  series, 
I,  2,  3,  .  .  .,  which  are  smaller  than  those  occurring 
in  K.  If  a  number  v  belongs  to  J,  all  numbers  less 
than  V  belong  to  J.  But  J  must  have  one  element 
i/j  such  that  J'l+i,  and  consequently  all  greater 
numbers,  do  not  belong  to  J,  because  otherwise 
J  would  contain  all  finite  numbers,  whereas  the 
numbers  belonging  to  K  are  not  contained  in  J. 
Thus  J  is  the  segment  {AbscJinitt)  (1,2,  3,  .  .  .,  j/j). 
The  number  v^-\-\=k^  is  necessarily  an  element  of 
K  and  smaller  than  the  rest. 

From  F  we  conclude  : 

G.  Every  aggregate  K={/c}  of  different  finite 
cardinal  numbers  can  be  brought  into  the  form  of 
a  series 

such  that 


K  =  (/Cp  /C2,  /C3, .  .  .) 

/Ci  *^  /C9  "^  '^Sj  •   •'  • 


§6 

The  Smallest  Transfinite  Cardinal  Number 

Aleph-Zero 

Aggregates  with  finite  cardinal  numbers  are  called 
''finite  aggregates,"  all  others  we  will  call  "trans- 
finite  aggregates "  and  their  cardinal  numbers 
*'  transfinite  cardinal  numbers." 

The  first  example  of  a  transfinite  aggregate  is 
given  by  the  totality  of  finite  cardinal  numbers  v  ; 


I04     THE  FOUNDING  OF  THE  THEORY 

we  call  its  cardinal  number  (§  i)  "  Aleph-zero  "  and 
denote  it  by  j^q  ;  thus  we  define 

(i)  5^0=  {i^}- 

That  j^o  is  a  transfinite  number,  that  is  to  say,  is 
not  equal  to  any  finite  number  /x,  follows  from  the 
simple  fact  that,  if  to  the  aggregate  {v}  is  added  a 
new  element  e^,  the  union-aggregate  ({i/},  e^  is 
equivalent  to  the  original  aggregate  {j^}.  For  we 
can  think  of  this  reciprocally  univocal  correspond- 
ence between  them  :  to  the  element  e^  of  the  first 
corresponds  the  element  i  of  the  second,  and  to  the 
element  v  of  the  first  corresponds  the  element  v-\-\  oi 
the  other.      By  §  3  we  thus  have 

(2)  ^?o+I=^^o• 

But  we  showed  in  §  5  that  //  -f-  i  is  always  different 
from  /x,  and  therefore  ^^  is  not  equal  to  any  finite 
number  fx. 

The  number  j^^  is  greater  than  any  finite  number  ix  : 

(3)  So>M- 

[493]    This    follows,    if   we  pay  attention    to   §   3, 
from  the  three  facts  that  />t  =  (i,  2,  3,  .  .  .,  /x),  that' 
no  part  of  the  aggregate  (i,  2,  3,  .  .  .,  ^)  is  equiva- 
lent to  the  aggregate  {v},  and  that  (i,  2,  3,  .  .  .,  yu) 
is  itself  a  part  of  {v}. 

On  the  other  hand,  h^^  is  the  least  transfinite 
cardinal  number.  If  a  is  any  transfinite  cardinal 
number  different  from  i^^,  then 

(4)  ^0<<'^' 


OF  TRANSFINITE  NUMBERS        105 

This  rests  on  the  following  theorems  : 

A.  Every  transfmite  aggregate  T  has  parts  with 
the  cardinal  number  ^^Q. 

Proof. — If,  by  any  rule,  we  have  taken  away  a 
finite  number  of  elements  t^,  t^^  .  .  .,/,,_i,  there 
always  remains  the  possibility  of  taking  away  a 
further  element  t^.  The  aggregate  {/,,},  where  v 
denotes  any  finite  cardinal  number,  is  a  part  of  T 
with  the  cardinal  number  t^^,  because  {/^}f\j{i'}  (§  i). 

B.  If  S  is  a  transfinite  aggregate  with  the  cardinal 
number  j^^,  and  S^  is  any  transfinite  part  of  S,  then 

Proof. — We  have  supposed  that  S  00  {v\.  Choose 
a  definite  law  of  correspondence  between  these  two 
aggregates,  and,  with  this  law,  denote  by  s^  that 
element  of  S  which  corresponds  to  the  element  v  of 
{i/},  so  that 

The  part  S^  of  S  consists  of  certain  elements  s^ 
of  S,  and  the  totality  of  numbers  k  forms  a  trans- 
finite part  K  of  the  aggregate  [v).  By  theorem  G 
of  §  5  the  aggregate  K  can  be  brought  into  the 
form  of  a  series 

where 

consequently  we  have 


io6     THE  FOUNDING  OF  THE   THEORY 

Hence  follows  that   Sj  oo  S,  and  therefore   Si  =  j^o- 
From  A  and  B  the  formula  (4)  results,  if  we  have 

regard  to  §  2. 

From  (2)  we  conclude,  by  adding  i  to  both  sides, 

and,  by  repeating  this 

(5)  ^*o  +  ^  =  «o- 
We  have  also 

(6)  «o  +  «o  =  ^*o- 

[494]  For,  by  (i)  of  §  3,  No  +  «o  is  the  cardinal  number 
({«J,  {by))  because 

Now,  obviously 

{.}=({2.-l},  {2.}), 
({2.-  I},   {2.})00(K},   {b^\ 

and  therefore 


The  equation  (6)  can  also  be  written 

t^o- 2  =  ^^0  5       . 
and,    by    adding    j^q    repeatedly  to   both  sides,   we 
find  that 

(7)  ^^o•^  =  I^•  «o  =  ^^o- 
We  also  have 

(8)  ^^o•No  =  ^^o• 


OF  TRANSFIX/TIL  NUAIBFKS         107 

Proof. — By   (6)   of    §    3,    Nq  •  t^o    ^-^    ^^^^    cardinal 
number  of  the  aggregate  of  bindings 

{{p.,  y)], 

where  ^  and  v  are  any  finite  cardinal  numbers  which 
are  independent  of  one  another.  If  also  X  repre- 
sents any  finite  cardinal  number,  so  that  {\},  (fj.), 
and  {v)  are  only  different  notations  for  the  same 
aggregate  o(  all  finite  numbers,  we  have  to  show 
that 

{(m,  ^)}fX^{X}. 

Let  us  denote  ^  +  1/  by  p\  then  ^  takes  all  the 
numerical  values  2,  3,  4,  .  .  .,  and  there  are  in  all 
p—  I  elements  (/x,  v)  for  which  ^-|-j/  =  p,  namely  : 

(1,^-1),    (2,p-2),..  ,    (p-I,   I). 

In  this  sequence  imagine  first  the  element  (i,  i), 
for  which  p=2,  put,  then  the  two  elements  for 
which  p=3,  then  the  three  elements  for  which 
p  =  4,  and  so  on.  Thus  we  get  all  the  elements 
(juL,  p)  in  a  simple  series  : 

(I,  i);(i,  2),(2,  i);(i,3),(2,  2),(3,  i);(i,4),(2,  3) 

and  here,  as  we  easily  see,  the  element  {ju,  v)  comes 
at  the  Xth  place,  where 

(9)  X  =  /x+  2 

The  variable  X  takes  every  numerical  value  i,  2,  3, 
.  .  .,   once.      Consequently,    by    means    of    (9),    a 


To8     THE  FOUNDING  OF  THE   THEORY 

reciprocally  univocal  relation   subsists  between  the 
aggregates  {v}  and  {(/x,  v)}. 

[495]  If  both  sides  of  the  equation  (8)  are  multi- 
plied by  j^o>  we  get  ^^^  = 'i:^^^  =  ^^^  and,  by  repeated 
multiplications  by  ^^^  we  get  the  equation,  valid 
for  every  finite  cardinal  number  v  : 

(lo)  ^^o"  =  «o• 

The  theorems  E  and  A  of  §  5  lead  to  this  theorem 
on  finite  aggregates  : 

C.  Every  finite  aggregate  E  is  such  that  it  is 
equivalent  to  none  of  its  parts. 

This  theorem  stands  sharply  opposed  to  the 
following  one  for  transfinite  aggregates  : 

D.  Every  transfinite  aggregate  T  is  such  that  it 
has  parts  T^  which  are  equivalent  to  it. 

Pi'oof. — By  theorem  A  of  this  paragraph  there  is 
a  part  S={^4  of  T  with  the  cardinal  number  «(,. 
Let  T  =  (S,  U),  so  that  U  is  composed  of  those 
elements  of  T  which  are  different  from  the  elements 
C  Let  us  put  Si  =  {/,+i},  Ti  =  (Si,  U)  ;  then  T^  is 
a  part  of  T,  and,  in  fact,  that  part  which  arises  out 
of  T  if  we  leave  out  the  single  element  t^.  Since 
S  00  Si,  by  theorem  B  of  this  paragraph,  and 
UooU,   we  have,   by   §    i,   T  rx;  T^. 

In  these  theorems  C  and  D  the  essential  differ- 
ence between  finite  and  transfinite  aggregates,  to 
which  I  referred  in  the  year  1877,  in  volume  Ixxxiv 
[1878]  of  Crelle's  Journal,  p.  242,  appears  in  the 
clearest  way. 

After   we    have   introduced   the   least   transfinite 


OF  TR A XS FINITE  NUMJUIRS        loo 

cardinal  number  Nq  and  derived  its  properties  tliat 
lie  the  most  readily  to  hand,  the  question  arises 
as  to  the  higher  cardinal  numbers  and  how  they 
proceed  from  h^^.  We  shall  show  that  the  trans- 
finite  cardinal  numbers  can  be  arranged  according 
to  their  magnitude,  and,  in  this  order,  form,  like 
the  finite  numbers,  a  '*  well-ordered  aggregate"  in 
an  extended  sense  of  the  words.  Out  of  ^^q  pro- 
ceeds, by  a  definite  law,  the  next  greater  cardinal 
number  j^^,  out  of  this  by  the  same  law  the  next 
greater  n^j  ^'^'^^  so  o^"*-  ^^t  even  the  unlimited 
sequence  of  cardinal  numbers 

No'  «i'  «2'  •  •  •)  «.',  ... 

does  not  exhaust  the  conception  of  transfinite 
cardinal  number.  We  will  prove  the  existence  of 
a  cardinal  number  which  we  denote  by  ^^^  and 
which  shows  itself  to  be  the  next  greater  to  all 
the  numbers  ^^  ;  out  of  it  proceeds  in  the  same 
way  as  i^^  out  of  ^?  a  next  greater  ^,^+1^  ^^^^  ^^  o^''» 
without  end. 

[496]  To  every  transhnite  cardinal  number  a 
there  is  a  next  greater  proceeding  out  of  it  accord- 
ing to  a  unitary  law,  and  also  to  every  unlimitedly 
ascending  well-ordered  aggregate  of  transfinite 
cardinal  numbers,  {a},  there  is  a  next  greater  pro- 
ceeding out  of  that  aggregate  in  a  unitary  way. 

For  the  rigorous  foundation  of  this  matter,  dis- 
covered in  1882  and  exposed  in  the  pamphlet 
Grundlagen  einer  allgemeitjen  MannicJifaltigkeits- 
lehre    (Leipzig,    1883)    and    in    volume    xxi    of   the 


110     THE  FOUNDING  OF  THE  THEORY 

MatJiematische  AnnaUn,  we  make  use  of  the  so- 
called  "ordinal  types "  whose  theory  we  have  to 
set  forth  in  the  following  paragraphs. 

§  7 

The  Ordinal  Types  of  Simply  Ordered 
Aggregates 

We  call  an  aggregate  M  "simply  ordered"  if  a 
definite  "order  of  precedence"  {Rmigordnung)  rules 
over  its  elements  m,  so  that,  of  every  two  elements 
m^  and  ?;^2>  ^^^  takes  the  "  lower  "  and  the  other  the 
' '  higher  "  rank,  and  so  that,  if  of  three  elements  ni^, 
jn^,  and  Wg,  nt-^,  say,  is  of  lower  rank  than  z//^,  and 
7n.2^  is  of  lower  rank  than  m^y  then  m^  is  of  lower 
rank  than  m^. 

The  relation  of  two  elements  ///j  and  ;;/^,  in  which 
m^  has  the  lower  rank  in  the  given  order  of  pre- 
cedence and  m^  the  higher,  is  expressed  by  the 
formulae  : 

(i)  m^  -<  m^,     in^  >-  m^ 

Thus,  for  example,  every  aggregate  P  of  points 
defined  on  a  straight  line  is  a  simply  ordered 
aggregate  if,  of  every  two  points  /^  and  p^  belong- 
ing to  it,  that  one  whose  co-ordinate  (an  origin  and 
a  positive  direction  having  been  fixed  upon)  is  the 
lesser  is  given  the  lower  rank. 

It  is  evident  that  one  and  the  same  aggregate  can 
be  "  simply  ordered  "  according  to  the  most  different 
laws.      Thus,  for  example,  witl^i   the  aggregate  R  of 


OF  TRANSFINITE  NUMBERS        \  1 1 

all  positive  rational  numbers//^  (where/  and  q  are 
relatively  prime  integers)  which  are  greater  than  o 
and  less  than  i,  there  is,  firstly,  their  "natural" 
order  according  to  magnitude  ;  then  they  can  be 
arranged  (and  in  this  order  we  will  denote  the 
aggregate  by  Rq)  so  that,  of  two  numbers /^/^^  and 
A/^-  for  which  the  sums  /i  +  ^i  and  p^  +  g^  have 
different  values,  that  number  for  which  the  corre- 
sponding sum  is  less  takes  the  lower  rank,  and,  if 
A  +  ^i=/2  +  ^2'  then  the  smaller  of  the  two  rational 
numbers  is  the  lower.  [497]  In  this  order  of 
precedence,  our  aggregate,  since  to  one  and  the 
same  value  oi  p-\-q  only  a  finite  number  of  rational 
numbers//^  belongs,  evidently  has  the  form 

-•^^0        \'l>   '2'   •   •    ■'       «"   •    •   '/        V2'    3'    4'    ."'    5  J    «»    5'    4'   •   •    VJ 

where 

r,  <  /-,+!. 

Always,  then,  when  we  speak  of  a  "simply 
ordered "  aggregate  M,  we  imagine  laid  down  a 
definite  order  or  precedence  of  its  elements,  in  the 
sense  explained  above. 

There  are  doubly,  triply,  j/-ply  and  a-ply  ordered 
aggregates,  but  for  the  present  we  will  not  consider 
them.  So  in  what  follows  we  will  use  the  shorter 
expression  "ordered  aggregate"  when  we  mean 
"simply  ordered  aggregate." 

Every  ordered  aggregate  M  has  a  definite  "ordinal 
type,"  or  more  shortly  a  definite  "type,"  which  we 
will  denote  by 

(2)  M. 


112     THE  FOUNDING  OF  THE  THEORY 

By  this  we  understand  the  general  concept  which 
results  from  M  if  we  only  abstract  from  the  nature 
of  the  elements  in^  and  retain  the  order  of  precedence 
among  them.  Thus  the  ordinal  type  M  is  itself  an 
ordered  aggregate  whose  elements  are  units  which 
have  the  same  order  of  precedence  amongst  one 
another  as  the  corresponding  elements  of  M,  from 
which  they  are  derived  by  abstraction. 

We  call  two  ordered  aggregates  M  and  N 
"similar"  {dhnlich)  if  they  can  be  put  into  a  bi- 
univocal  correspondence  with  one  another  in  such 
a  manner  that,  if  ?n-^  and  ni^  are  any  two  elements 
of  M  and  n^  and  n^  the  corresponding  elements  of  N, 
then  the  relation  of  rank  of  in^  to  m^  in  M  is  the 
same  as  that  of  n-^  to  n^  in  N.  Such  a  correspond- 
ence of  similar  aggregates  we  call  an  '*  imaging" 
{Abbildimg)  of  thes'e  aggregates  on  one  another.  In 
such  an  imaging,  to  every  part — which  obviously 
also  appears  as  an  ordered  aggregate — M^  of  M 
corresponds  a  similar  part  N^  of  N. 

We  express  the  similarity  of  two  ordered  aggre- 
gates M  and  N  by  the  formula  : 

(3)  MooN. 

Every  ordered  aggregate  is  similar  to  itself. 

If  two  ordered  aggregates  are  similar  to  a  third, 
they  are  similar  to  one  another. 

[498]  A  simple  consideration  shows  that  two 
ordered  aggregates  have  the  same  ordinal  type  if, 
and  only  if,  they  are  similar,  so  that,  of  the  two 
formula: 


OF  TRANSFINITE  NUMBERS        113 

(4)  M  =  N,     MOON, 

one  is  always  a  consequence  of  the  other. 

If,  with  an  ordinal  t}'pe  M  we  also  abstract  from 
the  order  of  precedence  of  the  elements,  we  get  (§  i) 
the  cardinal  number  M  of  the  ordered  aggregate  M, 
which  is,  at  the  same  time,  the  cardinal  number  of 
the  ordinal  type  M.  From  M  =  N  always  follows 
M  =  N,  that  is  to  say,  ordered  aggregates  of  equal 
types  always  have  the  same  power  or  cardinal 
number  ;  from  the  similarity  of  ordered  aggregates 
follows  their  equivalence.  On  the  other  hand,  two 
aggregates  may  be  equivalent  without  being  similar. 

We  will  use  the  small  letters  of  the  Greek  alphabet 
to  denote  ordinal  types.  If  a  is  an  ordinal  type, 
we  understand  by 

(5)  « 

its  corresponding  cardinal  number. 

The  ordinal  types  of  finite  ordered  aggregates 
offer  no  special  interest.  For  we  easily  convince 
ourselves  that,  for  one  and  the  same  finite  cardinal 
number  i/,  all  "simply  ordered  aggregates  are  similar 
to  one  another,  and  thus  have  one  and  the  same 
type.  Thus  the  finite  simple  ordinal  types  are 
subject  to  the  same  laws  as  the  finite  cardinal 
numbers,  and  it  is  allowable  to  use  the  same  signs 
I,  2,  3,  .  .  .,  Vy  .  .  .  for  them,  although  they  are 
conceptually  different  from  the  cardinal  numbers. 
The  case  is  quite  different  with  the  transfinite 
ordiiial   types  ;  for    to    one    and  the   same  cardinal 

8 


114     THE  FOUNDING  OF  THE   THEORY 

number  belong  innumerably  many  different  types  of 
simply  ordered  aggregates,  which,  in  their  totality, 
constitute  a  particular  ' '  class  of  types  "  (  Typenclasse). 
Every  one  of  these  classes  of  types  is,  therefore, 
determined  by  the  transfinite  cardinal  number  a 
which  is  common  to  all  the  types  belonging  to  the 
class.  Thus  we  call  it  for  short  the  class  of  types  [a]. 
That  class  which  naturally  presents  itself  first  to  us, 
and  whose  complete  investigation  must,  accordingly, 
be  the  next  special  aim  of  the  theory  of  transfinite 
aggregates,  is  the  class  of  types  [^^o]  which  embraces 
all  the  types  with  the  least  transfinite  cardinal 
number  ^^q.  From  the  cardinal  number  which 
determines  the  class  of  types  [a]  we  have  to  dis- 
tinguish that  cardinal  number  a'  which  for  its  part 
[499]  ^^  determmed  by  the  class  of  types  [a].  The 
latter  is  the  cardinal  number  which  (§  i)  the  class 
[a]  has,  in  so  far  as  it  represents  a  well-defined 
aggregate  whose  elements  are  all  the  types  a  with 
the  cardinal  number  a.  We  will  see  that  a'  is 
different  from  a,  and  indeed  always  greater  than  a. 

If  in  an  ordered  aggregate  M  all  the  relations  of 
precedence  of  its  elements  are  inverted,  so  that 
' '  lower  "  becomes  ' '  higher  "  and  ' '  higher  "  becomes 
' '  lower "  everywhere,  we  again  get  an  ordered 
aggregate,  which  we  will  denote  by 

(6)  *M 

and    call    the    "inverse"    of   M.       We    denote  the 
ordinal  type  of  *M,  if  a=  M,  by 

(7) 


OF  TRANSFINITE  NUMBERS        115 

It  may  happen  that  *a  =  a,  as,  for  example,  in  the 
case  of  finite  types  or  in  that  of  the  type  of  the 
aggregate  of  all  rational  numbers  which  are  greater 
than  o  and  less  than  i  in  their  natural  order  of 
precedence.  This  type  we  will  investigate  under 
the  notation  ;/. 

We  remark  further  that  two  similarly  ordered 
aggregates  can  be  imaged  on  one  another  either  in 
one  manner  or  in  many  manners  ;  in  the  first  case 
the  type  in  question  is  similar  to  itself  in  only  one 
way,  in  the  second  case  in  many  ways.  Not  only 
all  finite  types,  but  the  types  of  transfinite  "well- 
ordered  aggregates,"  which  will  occupy  us  later 
and  which  we  call  transfinite  "ordinal  numbers," 
are  such  that  they  allow  only  a  single  imaging  on 
themselves.  On  the  other  hand,  the  type  ri  is 
similar  to  itself  in  an  infinity  of  ways. 

We  will  make  this  difference  clear  by  two  simple 
examples.  By  o)  we  understand  the  type  of  a  well- 
ordered  aggregate 

in  which 

and  where  p  represents  all  finite  cardinal  numbers  in 
turn.      Another  well-ordered  aggregate 

with  the  condition 

of  the  same  type  od  can   obviously  only  be   imaged 


Ti6     THE  FOUNDING  OF  THE   THEORY 

on  the  former  in  such  a  way  that  e^,  and  /,  are 
correspondhig  elements.  For  e^,  the  lowest  element 
in  rank  of  the  first,  must,  in  the  process  of  imaging, 
be  correlated  to  the  lowest  element /^  of  the  second, 
the  next  after  e^  in  rank  {e^)  to/g,  the  next  after/^, 
and  so  on.  [500]  Every  other  bi-univocal  corre- 
spondence of  the  two  equivalent  aggregates  {e^}  and 
{/,}  is  not  an  "imaging"  in  the  sense  which  we 
have  fixed  above  for  the  theory  of  types. 

On    the    other    hand,    let    us    take    an     ordered 
aggregate  of  the  form 

where  v  represents  all  positive  and  negative  finite 
integers-,  including  o,  and  where  likewise 

This  aggregate  has  no  lowest  and  no  highest 
element  in  rank.  Its  type  is,  by  the  definition  of 
a  sum  given  in  §  8, 

It  is  similar  to  itself  in  an  infinity  of  ways.  For 
let  us  consider  an  aggregate  of  the  same  type 

where 

Then  the  two  ordered  aggregates  can  be  so  imaged 
on  one  another  that,  if  we  understand  by  vq  a 
definite  one  of  the  numbers  /,  to  the  element  e,'  of 


OF  TRANSFINITE  NUMBERS         ti; 

the  first  the  element /j,^,'_^_,/  of  the  second  corresponds. 
Since  j/q'  is  arbitrary,  we  liave  here  an  infinity  of 
imagings. 

The  concept  of  ''ordinal  type"  developed  here, 
when  it  is  transferred  in  like  manner  to  "multiply 
ordered  aggregates,"  embraces,  in  conjunction  with 
the  concept  of  "cardinal  number"  or  "power" 
introduced  in  §  i,  everything  capable  of  being 
numbered  {Anzahlmdssige)  that  is  thinkable,  and 
in  this  sense  cannot  be  further  generalized.  It 
contains  nothing  arbitrary,  but  is  the  natural  ex- 
tension of  the  concept  of  number.  It  deserves  to 
be  especially  emphasized  that  the  criterion  of 
equality  (4)  follows  with  absolute  necessity  from 
the  concept  of  ordinal  type  and  consequently 
permits  of  no  alteration.  The  chief  cause  of  the 
grave  errors  in  G.  Veronese's  Gnuidziige  der 
Geoinetrie  (German  by  A.  Schepp,  Leipzig,  1894) 
is  the  non-recognition  of  this  point. 

On  page  30  the  ' '  number  {Anrjahl  oder  ZaJd) 
of  an  ordered  group  "  is  defined  in  exactly  the  same 
way  as  what  we  have  called  the  "ordinal  type  of 
a  simply  ordered  aggregate "  {Ztir  Lehre  vorn 
Transfiniten,  Halle,  1890,  pp.  68-75  i  reprinted 
from  the  'ZeitscJir.  fiir  Pliilos.  und  philos.  Kritik 
for  1887).  [501]  But  Veronese  thinks  that  he 
must  make  an  addition  to  the  criterion  of  equality. 
He  says  on  page  31:  "Numbers  whose  units 
correspond  to  one  another  uniquely  and  in  the 
same  order  and  of  which  the  one  is  neither  a  part 
of  the  other  nor  equal  to  a  part  of  the  other  are 


ii8     THE  FOUNDING  OF  THE   THEORY 

equal. "  *  This  definition  of  equality  contains  a 
circle  and  thus  is  meaningless.  For  what  is 
the  meaning  of  "not  equal  to  a  part  of  the 
other"  in  this  addition?  To  answer  this  question, 
we  must  first  know  when  two  numbers  are  equal 
or  unequal.  Thus,  apart  from  the  arbitrariness 
of  his  definition  of  equality,  it  presupposes  a 
definition  of  equality,  and  this  again  presupposes 
a  definition  of  equality,  in  which  we  must  know 
again  what  equal  and  unequal  are,  and  so  on  ad 
infinitum.  After  Veronese  has,  so  to  speak,  given 
up  of  his  own  free  will  the  indispensable  foundation 
for  the  comparison  of  numbers,  we  ought  not  to 
be  surprised  at  the  lawlessness  with  which,  later 
on,  he  operates  with  his  pseudo-transfinite  numbers, 
and  ascribes  properties  to  them  which  they  cannot 
possess  simply  because  they  themselves,  in  the 
form  imagined  by  him,  have  no  existence  except 
on  paper.  Thus,  too,  the  striking  similarity  of  his 
' '  numbers  "  to  the  very  absurd  ' '  infinite  numbers  " 
in  Fontenelle's  Geo7netrie  de  IFnfifii  (Paris,  1727) 
becomes  comprehensible.  Recently,  W.  Killing 
has  given  welcome  expression  to  his  doubts  con- 
cerning the  foundation  of  Veronese's  book  in  the 
Index  lectionuni  of  the  Miinster  Academy  for  1895- 
i896.t 

*  In  the  original  Italian  edition  (p.  27)  this  passage  runs  :  "  Numeri 
le  unita  dei  quali  si  corrispondono  univocamente  e  nel  medesimo  ordine, 
e  di  cui  1'  uno  non  e  parte  o  uguale  ad  una  parte  dell'  altro,  sono  uguali." 

t  [Veronese  replied  to  this  in  Math.  Ann.,  vol.  xlvii,  1897,  pp.  423- 
432.      Cf.  Killing,  ibid.,  vol.  xlviii,  1897,  pp.  425-432.] 


OF  TRANSFINITE  NUMBERS        119 


Addition  and  Multiplication  of  Ordinal  Types 

The  union-aggregate  (M,  N)  of  two  aggregates 
M  and  N  can,  if  M  and  N  are  ordered,  be  conceived 
as  an  ordered  aggregate  in  which  the  relations  of 
precedence  of  the  elements  of  M  among  themselves 
as  well  as  the  relations  of  precedence  of  the  elements 
of  N  among  themselves  remain  the  same  as  in  M 
or  N  respectively,  and  all  elements  of  M  have  a 
lower  rank  than  all  the  elements  of  N.  If  M'  and 
N'  are  two  other  ordered  aggregates,  M  Oo  M'  and 
N  fV)  N',  [502]  then  (M,  N)  cx>  (M^  N')  ;  so  the 
ordinal  type  of  (M,  N)  depends  only  on  the  ordinal 
types  M  =  a  and  N  =  /3.      Thus,  we  define: 


(1)  a  +  /3  =  (M,  N). 

In  the  sum  ct  +  /3  we  call  a  the  ''augend  "  and  /3  the 
"addend." 

For  any  three  types  we  easily  prove  the  associa- 
tive law  : 

(2)  a  +  (/3  +  y)=:(a4./3)  +  y. 

On  the  other  hand,  the  commutative  law  is  not 
valid,  in  general,  for  the  addition  of  types.  We 
see  this  by  the  following  simple  example. 

If  o)  is   the  type,    already   mentioned   in   §   7,    of 
the  well-ordered  aggregate 


I20     THE  FOUNDING  OF  THE   THEORY 

then  I  +w  is  not  equal  to  (o+  i.      For,  if/ is  a  new 
element,  we  have  by  (i)  : 


l+a)  =  (/E), 
a,+  l=(E;7)- 
But  the  aggregate 

(/E)=(y;^i, ^2.  •  •-^.'•- •) 

is  similar  to  the  aggregate  E,  and  consequently 

I  +co  =  a). 

On  the  contrary,  the  aggregates  E  and  (E,  f)  are 
not  similar,  because  the  first  has  no  term  which  is 
highest  in  rank,  but  the  second  has  the  highest 
term /!      Thus  coH-  i  is  different  from  a)=  i  +a). 

Out  of  two  ordered  aggregates  M  and  N  with 
the  types  a  and  /3  we  can  set  up  an  ordered 
aggregate  S  by  substituting  for  every'  element  n  of 
N  an  ordered  aggregate  M,,  which  has  the  same 
type  a  as  M,  so  that 

(3)  M,  =  a; 
and,  for  the  order  of  precedence  in 

(4)  S  =  {M„} 

we  make  the  two  rules  : 

(i)  Every  two  elements  of  S  which  belong  to 
one  and  the  same  aggregate  M^,  are  to  retain  in 
S  the  same  order  of  precedence  as  in  M,,  ; 

(2)  Every  two  elements  of  S  which  belong  to  two 
different  aggregates  M^^  and  M^,.^  have  the  same 
relation  of  precedence  as  n^  and  71^  have  in  N. 


OF  TRANSFfNITE  NUMBERS        121 

The  ordinal  type  of  S  depends,  as  we  easily  see, 
only  on  the  types  a  and  /5  ;  we  define 

(5)  «./3  =  S. 

[503]  In  this  product  a  is  called  the  "  multiplicand  " 
and  /3  the  "  multiplier." 

In  any  definite  imaging  of  M  on  M„  let  in^  be  the 
element  of  M„  that  corresponds  to  the  element  m 
of  M;  we  can  then  also  write 

(6)  S  =  {/«„}. 

Consider   a    third   ordered   aggregate   V  =  [p]   with 
the  ordinal  type  P  =  y,  then,  by  (5), 


a.(/5-y)  =  {^^')}. 
But  the  two  ordered  aggregates  {(m,,)  }  and  {^^^(n.)} 
are  similar,  and  are  imaged  on  one  another  if  we 
regard  the  elements  (^z^^)  ^^^^  ?''^/«.\  as  correspond- 
ing. Consequently,  for  three  types  a,  ^8,  and  y 
the  associative  law 

(7)  (a.^).y  =  a.(/5.y) 

subsists.      From  (i)  and  (5)  follows  easily  the  dis- 
tributive law 

(8)  a.(^-hy)  =  a./3-{-a.y; 

but  only  in  this  form,   where  the   factor  with  two 
terms  is  the  multiplier. 

On  the  contrary,    in  the  multiplication  of  types 
as   in  their    addition,   the    commutative  law  is  not 


\ 


122     THE  FOUNDING  OF  THE   THEORY 

generally  valid.      For   example,    2.0)    and  w  .2    are 
different  types  ;  for,  by  (5), 


while 

W.    2  =  {e^,    ^2'   •    •    •>    ^.',   ...;    /l,/2,   ••.,/',-.    •) 

is  obviously  different  from  o). 

If  we  compare  the  definitions  of  the  elementary 
operations  for  cardinal  numbers,  given  in  §  3,  with 
those  established  here  for  ordinal  types,  we  easily 
see  that  the  cardinal  number  of  the  sum  of  two 
types  is  equal  to  the  sum  of  the  cardinal  numbers 
of  the  single  types,  and  that  the  cardinal  number 
of  the  product  of  two  types  is  equal  to  the  pro- 
duct of  the  cardinal  numbers  of  the  single  types. 
Every  equation  between  ordinal  types  which  pro- 
ceeds from  the  two  elementary  operations  remains 
correct,  therefore,  if  we  replace  in  it  all  the  types 
by  their  cardinal  numbers. 

[504]  §  9 

The  Ordinal  Type  r]  of  the  Aggregate  R  of  all 
Rational  Numbers  which  are  Greater  than 
o  and  Smaller  than  i,  in  their  Natural 
Order  of  Precedence 

By  R  we  understand,  as  in  §  7,  the  system  of 
all  rational  numbers  p\q  {p  and  q  being  relatively 
prime)  which  >o  and  <  i,  in  their  natural  order 
of  precedence,    where  the  magnitude  of  a  number 


} 


/ 


OF  TRANSFINITR  NUMBERS        123 

determines  its  rank.  We  denote  the  ordinal  t)'pe 
of  R  by  >y  : 

(1)  >/  =  R. 

But  we  have  put  the  same  aggregate  in  another 
order  of  precedence  in  which  we  call  it  Rq.  This 
order  is  determined,  in  the  first  place,  by  the 
magnitude  of  p-\-q,  and  in  the  second  place — for 
rational  numbers  for  which  p -\- q  has  the  same  value 
— by  the  magnitude  of  pjq  itself.  The  aggregate 
Rq  is  a  well-ordered  aggregate  of  type  w  : 

(2)  Ro  =  (>'i,  ^'2,  .  .  .,  i^.M  .  •  •),     where     r,<r,+i, 

(3)  Ro  =  ^- 

Both  R  and  Rq  have  the  same  cardinal  number 
since  they  only  differ  in  the  order  of  precedence 
of  their    elements,    and,    since   we   obviously   have 

Ro  =  j^o'  ^^'^  ^^^°  \\di\^ 

(4)  R  =  7/  =  No- 

Thus  the  type  ri  belongs  to  the  class  of  types  [^^o]. 

Secondly,  we  remark  that  in  R  there  is  neither 
an  element  which  is  lowest  in  rank  nor  one  which 
is  highest  in  rank.  Thirdly,  R  has  the  property 
that  between  every  two  of  its  elements  others  lie. 
This  property  we  express  by  the  words  :  R  is 
"everywhere  dense"  {uberalldicht), 

VVe  will  now  show  that  these  three  properties 
characterize  the  type  >/  of  R,  so  that  we  have  the 
following  theorem  : 


124     THE  FOUNDING   OF  THE   THEORY 

If  we  have  a  simply  ordered  aggregate  M  such 
that     _ 

{a)  M  =  «o; 

{b)  M  has  no  element  which  is  lowest   in  rank, 

and  no  highest  ; 
{c)  M  is  everywhere  dense  ; 
then  the  ordinal  type  of  M  is  ^  : 

Pjvof. — Because  of  the  condition  {a),  M  can  be 
brought  into  the  form  [505]  ^^  ^  well-ordered 
aggregate  of  type  w  ;  having  fixed  upon  such  a 
form,  we  denote  it  by  Mq  and  put 

(5)  Mo  =  (?//i,  Wg,  .  .  .,  ;//„,  .  .  .). 
We  have  now  to  show  that 

(6)  MooR; 

that  is  to  say,  we  must  prove  that  M  can  be  imaged 
on  R  in  such  a  way  that  the  relation  of  precedence 
of  any  and  every  two  elements  in  M  is  the  same 
as  that  of  the  two  corresponding  elements  in  R. 

Let  the  element  7\  in  R  be  correlated  to  the 
element  m^  in  M.  The  element  rg  h^s  a  definite 
relation  of  precedence  to  7\  in  R.  Because  of  the 
condition  {b),  there  are  infinitely  many  elements 
vi^,  of  M  which  have  the  same  relation  of  precedence 
in  M  to  m-^  as  r^^  to  7\  in  R  ;  of  them  we  choose 
that  one  which  has  the  smallest  index  in  M^,  let  it 
be  ifii  and  correlate  it   to  r^.      The  element  r,  has 


in  R  definite  relations  of  precedence  to  i\  and  r^  ; 
because    of   the   conditions  (b)  and  (c)  there  is  an 


OF  TRANSFINITIi  NUMBERS        125 

infinity  of  elements  in^,  of  M  which  have  the  same 

relation  of  precedence  to  m^  and  nii  in  M  as  rg  to  }\ 

and  r.y  to  R  ;  of  them  we  choose  that — let  it  be  ;;/, 
z  5 

— which  has  the  smallest  index  in  Mq,  and  correlate 
it  to  Tg.  According  to  this  law  we  imagine  the 
process  of  correlation  continued.  If  to  the  v 
elements 

^1'    '''2'    ^3)    •    •    •  )   ^i' 

of  R  are  correlated,  as  images,  definite  elements 

m^,  nu,  in^,  .  .  .,  in, 

which  have  the  same  relations  of  precedence  amongst 
one  another  in  M  as  the  corresponding  elements  in 
R,  then  to  the  element  ;v+i  of  R  is  to  be  correlated 
that  element  ni,^  of  M  which  has  the  smallest 
index  in  Mq  of  those  which  have  the  same  relations 
of  precedence  to 

;;/i,  m,^,  in,^,  .  .  . ,  m,^ 

in  M  as  r^+i  to  r^,  r^,  .  .  .,  r^  in  R. 

In  this  manner  we  have  correlated  definite 
elements  m.  of  M  to  all  the  elements  }\  of  R,  and 
the  elements  m,^  have  in  M  the  same  order  of  pre- 
cedence as  the  corresponding  elements  i\  in  R.  But 
we  have  still  to  show  that  the  elements  m,^  include 
all  the  elements  in^  of  M,  or,  what  is  the  same 
thing,  that  the  series 

I  )    '2'   '3'    •    •    •  '   ^f '   •    •    • 

[506]  is  only  a  permutation  of  the  series 
I,  2,  3,  ....',..  . 


126     THE  FOUNDING  OF  THE  THEORY 

We  prove  this  by  a  complete  induction  :  we  will 
show  that,  if  the  elements  m^,  m^,  .  .  .,  m^  appear 
in  the  imaging,  that  is  also  the  case  with  the 
following  element  m^+i. 

Let  X  be  so  great  that,  among  the  elements 

w,,  nil,  nil ,  .  .  .,  lUu, 

■■■as  '^ 

the  elements 

m^,  in,,  .  .  .,  ;;/,, 

which,  by  supposition,  appear  in  the  imaging,  are 
contained.  It  may  be  that  also  ?;/,,+i  is  found 
among  them  ;  then  /;2^+i  appears  m  the  imaging. 
But  if  /;^^+i  is  not  among  the  elements 

m^,  in,,  m,,  .  .  .,  vi,^, 

then  ///^+i  has  with  respect  to  these  elements  a 
definite  ordinal  position  in  M  ;  infinitely  many 
elements  in  R  have  the  same  ordinal  position  in  R 
with  respect  to  i\,  rg,  .  .  . ,  r^,  amongst  which  let 
/'x+a  be  that  with  the  least  index  in  Rq.  Then  ni^^i 
has,  as  we  can  easily  make  sure,  the  same  ordinal 
position  with  respect  to 

m^,  m,,  m,,  .  .  .,  nh^^^_^ 
in  M  as  r^j^„  has  with  respect  to 

^'d    ''2)    •     •    •)    ^^A  +  a--l 

in  R.  Since  in^,  in.^^,  •  •  • ,  ^f^u  have  already  appeared 
in  the  imaging,  ni^^i  is  that  element  with  the  smallest 
index  in  M  which  has  this  ordinal  position  with 
respect  to 


OF  TRANSFJNITE  NUMBERS        127 
Consequently,  according  to  our  law  of  correlation, 


''^^,+^  =  '^'^+1- 


Thus,  in  this  case  too,  the  element  ;;/^.|.i  appears  in 
the  imaging,  and  r^+^  is  the  element  of  R  which  is 
correlated  to  it. 

We  see,  then,  that  by  our  manner  of  correlation, 
the  whole  aggregate  M  is  imaged  on  the  whole 
aggregate  R  ;  M  and  R  are  similar  aggregates, 
which  was  to  be  proved. 

From  the  theorem  which  we  have  just  proved 
result,  for  example,  the  following  theorems  : 

[507]  The  ordinal  type  of  the  aggregate  of  all 
negative  and  positive  rational  numbers,  including 
zero,  in  their  natural  order  of  precedence,  is  r^. 

The  ordinal  type  of  the  aggregate  of  all  rational 
numbers  which  are  greater  than  a  and  less  than  b, 
in  their  natural  order  of  precedence,  where  a  and  b 
are  any  real  numbers,  and  a  <b,  is  rj. 

The  ordinal  type  of  the  aggregate  of  all  real  alge- 
braic numbers  in  their  natural  order  of  precedence  is  tj. 

The  ordinal  type  of  the  aggregate  of  all  real  alge- 
braic numbers  which  are  greater  than  a  and  less 
than  b,  in  their  natural  order  of  precedence,  where 
a  and  b  are  any  real  numbers  and  a<b,  is  ;/. 

P^or  all  these  ordered  aggregates  satisfy  the  three 
conditions  required  in  our  theorem  for  M  (see 
CygWq' s  /ou7^nal,  vol.  Ixxvii,  p.  258).* 

If  we  consider,  further,  aggregates  with  the  types 
— according  to  the  definitions  given  in  §  8 — written 

[*   C/.  Section  V  of  the  Introduction.] 


128     THE  FOUNDING  OF  THE   THEORY 

'/  +  ';,  nn^  (i+>7)'7,  ('?+i)'7.  (!+>?+ i>/,  we  find  that 
those  three  conditions  are  also  fulfilled  with  them. 
Thus  we  have  the  theorems  : 


(7) 

>;  +  ')  =  '?, 

(8) 

m  =  i> 

(9) 

(I  +,,),,  =  ,,, 

(10) 

(>;+  !)>;  =  >;, 

(11) 

(l+,,+  l)ri  =  >i. 

The  repeated  application  of  (7)  and  (8)  gives  for 
every  finite  number  u  : 

(12)  t1'V  =  t1, 

(13)  n"  =  ^- 

On  the  other  hand  we  easily  see  that,  for  j/>  I,  the 
types  1+^,  i]-^i,  v.r],  i+>?+i  are  different  both 
from  one  another  and  from  t].      We  have 

(14)  ;^+I+,y  =  ^, 

but  r]  +  i/  +  t],  for  ^/>  I,  is  different  from  rj. 
Finally,  it  deserves  to  be  emphasized  that 

(15)  *>;  =  ';. 

[508]  §  10 

The  Fundamental  Series  contained  in  a 
Transfinite  Ordered  Aggregate 

Let  us  consider  any  simply  ordered  transfinite 
aggregate  M.  Every  part  of  M  is  itself  an  ordered 
aggregate.      For    the  study  of  the   type  M,   those 


OF  TRANSFINITE  NUMBERS        129 

parts  of  M  which  have  the  types  w  and  *a)  appear  to 
be  especially  valuable  ;  we  call  them  "  fundamental 
series  of  the  first  order  contained  in  M,"  and  the 
former — of  type  w — we  call  an  ''ascending"  series, 
the  latter — of  type  *o) — a  * '  descending  "  one.  Since 
we  limit  ourselves  to  the  consideration  of  funda- 
mental series  of  the  first  order  (in  later  investiga- 
tions fundamental  series  of  higher  order  will  also 
occupy  us),  we  will  here  simply  call  them  ''funda- 
mental series."  Thus  an  "ascending  fundamental 
series  "  is  of  the  form 

(i)  {a,},      where     ^,<^,+i; 

a  "  descending  fundamental  series  "  is  of  the  form 

(2)  {b^},     where     b^)^  b^+i. 

The  letter  v,  as  well  as  /c,  X,  and  /x,  has  everywhere 
in  our  considerations  the  signification  of  an  arbitrary 
finite  cardinal  number  or  of  a  finite  type  (a  finite 
ordinal  number). 

We  call  two  ascending  fundamental  series  {a^}  and 
{a\)  in  M  "coherent"  {zusammengehbrig),  in  signs 

(3)  {^.}  11  {<}) 

if,  for  every  element  a^  there  are  elements  a\  such 
that 

and  also  for  every  element  a\  there  are  elements  a^ 
such  that 

9 


I30     THE  FOUNDING  OF  THE   THEORY 

Two  descending  fundamental  series  {by}  and  {b' ^ 
in  M  are  said  to  be  "coherent,"  in  signs 

(4)  {K)  II  {b'.\. 

if  for  every  element  b^  there  are  elements  b\  such 
that 

b.  >  b\, 

and  for  every  element  b\  there  are  elements  b^j,  such 
that 

K  >  b,. 

An  ascending  fundamental  series  {a^}  and  a 
descending  one  {b^}  are  said  to  be  "coherent,"  in 
signs 

[509]    (5)  WlllW, 

if  (a)  for  all  values  of  v  and  /n 

and  (<^)  in  M  exists  at  most  one  (thus  either  only 
one  or  none  at  all)  element  m^^  such  that,  for  all  i/'s, 

^.  <^  f^h  <  ^>" 

Then  we  have  the  theorems  : 

A.  If  two  fundamental  series  are  coherent  to  a 
third,  they  are  also  coherent  to  one  another. 

B.  Two  fundamental  series  proceeding  in  the 
same  direction  of  which  one  is  part  of  the  other  are 
coherent. 

If  there  exists   in   M  an  element    m^   which   has 


OF  TRANSFINITE  NUMBERS        131 

such  a  position  with  respect  to  the  ascending  funda- 
mental series  [a^]  that  : 
{a)  for  every  v 

{b)  for  every  element  m  of  M  that  precedes  m^ 
there  exists  a  certain  number  vq  such  that 

a„  >^  m,     for     v'^Vft, 

then  we  will  call  iUq  a  "limiting  element  (Grenz- 
element)  of  {^J  in  M  "  and  also  a  "  principal  element 
{Haupt element)  of  M."  In  the  same  way  we  call 
niQ  a  "principal  element  of  M  "  and  also  "  Hmiting 
element  of  [b^  in  M"  if  these  conditions  are 
satisfied  : 

{a)  for  every  v 

ip)  for  every  element  m  of  M  that  follows  m^ 
exists  a  certain  number  v^  such  that 

b^  >^  m,     for     v^Vf^, 

A  fundamental  series  can  never  have  more  than 
one  limiting  element  in  M  ;  but  M  has,  in  general, 
many  principal  elements. 

We  perceive  the  truth  of  the  following  theorems  : 

C.  If  a  fundamental  series  has  a  limiting  element 
in  M,  all  fundamental  series  coherent  to  it  have  the 
same  limiting  element  in  M. 

D.  If  two  fundamental  series  (whether  proceeding 
in  the  same  or  in  opposite  directions)  have  one  and 
the  same  limiting  element  in  M,  they  are  coherent. 


132     THE  FOUNDING  OF  THE   THEORY 

if  M  and  M'  are  two  similarly  ordered  aggregates, 
so  that 

(6)  M  =  M', 

and  we  fix  upon  any  imaging  of  the  two  aggregates, 
then  we  easily  see  that  the  following  theorems 
hold: 

[510]  E.  To  every  fundamental  series  in  M 
corresponds  as  image  a  fundamental  series  in  M', 
and  inversely  ;  to  every  ascending  series  an  ascending 
one,  and  to  every  descending  series  a  descending 
one  ;  to  coherent  fundamental  series  in  M  corre- 
spond as  images  coherent  fundamental  series  in  M', 
and  inversely. 

F.  If  to  a  fundamental  series  in  M  belongs  a 
limiting  element  in  M,  then  to  the  corresponding 
fundamental  series  in  M'  belongs  a  limiting  element 
in  M',  and  inversely  ;  and  these  two  limiting 
elements  are  images  of  one  another  in  the  imaging. 

G.  To  the  principal  elements  of  M  correspond  as 
images  principal  elements  of  M',  and  inversely. 

If  an  aggregate  M  consists  of  principal  elements, 
so  that  every  one  of  its  elements  is  a  principal 
element,  we  call  it  an  ''aggregate  which  is  dense 
in  itself  {insichdichte  Mengey  If  to  every  funda- 
mental series  in  M  there  is  a  limiting  element  in  M, 
we  call  M  a  "closed  {abgeschlossene)  aggregate." 
An  aggregate  which  is  both  "dense  in  itself"  and 
"closed"  is  called  a  "perfect  aggregate."  If  an 
aggregate  has  one  of  these  three  predicates,  every 
similar    aggregate    has    the    same    predicate  ;    thus 


OF  TRANSFINITE  NUMBERS        133 

these  predicates  can  also  be  ascribed  to  the  corre- 
sponding ordinal  types,  and  so  there  are  "types 
which  are  dense  in  themselves,"  "closed  types," 
"perfect  types,"  and  also  "everywhere-dense 
types  "  (§  9). 

For  example,  >;  is  a  type  which  is  "dense  in 
itself, "  and,  as  we  showed  in  §  9,  it  is  also  ' '  every- 
where-dense," but  it  is  not  "closed."  The  types 
ft)  and  *ft)  have  no  principal  elements,  but  w-\-v  and 
i/  +  *ft)  each  have  a  principal  element,  and  are 
"closed"  types.  The  type  0^.3  has  two  principal 
elements,  but  is  not  "closed";  the  type  w.3-f^ 
has  three  principal  elements,  and  is  "closed." 

§11 

The  Ordinal  Type  Q  of  the  Linear 

Continuum  X 

We  turn  to  the  investigation  of  the  ordinal  type 
of  the  aggregate  X=  [x]  of  all  real  numbers  x,  such 
that  x>^o  and  <  i,  in  their  natural  order  of  pre- 
cedence, so  that,  with  any  two  of   its   elements  x 

and  x\ 

x^x\      if    x<x'. 

Let  the  notation  for  this  type  be 

(I)  x  =  e. 

[511]  From  the  elements  of  the  theory  of  rational 
and  irrational  numbers  we  know  that  every  funda- 
mental series  {x^}  in  X  has  a  limiting  element  Xq  in 
X,   and  that  also,  inversely,  every  element  x  of  X 


134     THE  FOUNDING  OF  THE   THEORY 

is  a  limiting  element  of  coherent  fundamental  series 
in  X.  Consequently  X  is  a  "perfect  aggregate" 
and  0  is  a  "perfect  type." 

But  Q  is  not  sufficiently  characterized  by  that  ; 
besides  that  we  must  fix  our  attention  on  the 
following  property  of  X.  The  aggregate  X  contains 
as  part  the  aggregate  R  of  ordinal  type  >]  investi- 
gated in  §  9,  and  in  such  a  way  that,  between  any 
two  elements  x^  and  x^  of  X,  elements  of  R  lie. 

We  will  now  show  that  these  properties,  taken 
together,  characterize  the  ordinal  type  Q  of  the  linear 
continuum  X  in  an  exhaustive  manner,  so  that  we 
have  the  theorem  : 

If  an  ordered  aggregate  M  is  such  that  (a)  it  is 
"perfect,"  and  {b)  in  it  is  contained  an  aggregate  S 
with  the  cardinal  number  S  =  h?o  ^^^  which  bears 
such  a  relation  to  M  that,  between  any  two  elements 
Wq  and  m^  of  M  elements  of  S  lie,  then  M  =  0. 

Proof. — If  S  had  a  lowest  or  a  highest  element, 
these  elements,  by  {b),  would  bear  the  same  character 
as  elements  of  M  ;  we  could  remove  them  from  S 
without  S  losing  thereby  the  relation  to  M  ex- 
pressed in  {b).  Thus,  we  suppose  that  S  is  without 
lowest  or  highest  element,  so  that,  by  §  9,  it  has 
the  ordinal  type  t].  ¥or  since  S  is  a  part  of  M, 
between  any  two  elements  Sq  and  s^  of  S  other 
elements  of  S  must,  by  {b),  lie.  Besides,  by  {b)  we 
have  S  =  «o'  Thus  the  aggregates  S  and  R  are 
"similar"  to  one  another. 

(2)  S  ro  R. 


OF  TRANSFINITE  NUMBERS         135 

We  fix  on  any  "imaging"  of  R  on  S,  and  assert 
that  it  gives  a  definite  "  imaging  "  of  X  on  M  in  the 
following  manner  : 

Let  all  elements  of  X  which,  at  the  same  time, 
belong  to  the  aggregate  R  correspond  as  images  to 
those  elements  of  M  which  are,  at  the  same  time, 
elements  of  S  and,  in  the  supposed  imaging  of 
R  on  S,  correspond  to  the  said  elements  of  R. 
But  li  Xq  is  an  element  of  X  which  does  not  belong 
to  R,  Xq  may  be  regarded  as  a  limiting  element  of 
a  fundamental  series  {x^]  contained  in  X,  and  this 
series  can  be  replaced  by  a  coherent  fundamental 
series  {r^J  contained  in  R.  To  this  [512]  corre- 
sponds as  image  a  fundamental  series  [s-^^]  in  S  and 
M,  which,  because  of  {a),  is  limited  by  an  element 
niQ  of  M  that  does  not  belong  to  S  (F,  §  10).  Let 
this  element  m^  of  M  (which  remains  the  same,  by 
E,  C,  and  D  of  §  10,  if  the  fundamental  series 
[x^}  and  [r^^]  are  replaced  by  others  limited  by  the 
same  element  x^  in  X)  be  the  image  of  x^^  in  X. 
Inversely,  to  every  element  m^  of  M  which  does  not 
occur  in  S  belongs  a  quite  definite  element  x^  of  X 
which  does  not  belong  to  R  and  of  which  m^  is  the 
image. 

In  this  manner  a  bi-univocal  correspondence 
between  X  and  M  is  set  up,  and  we  have  now 
to  show  that  it  gives  an  "imaging"  of  these 
aggregates. 

This  is,  of  course,  the  case  for  those  elements  of 
X  which  belong  to  R,  and  for  those  elements  of  M 


136  TRANSFINITE  NUMBERS 

which  belong  to  S.  Let  us  compare  an  element  r 
of  R  with  an  element  x^  of  X  which  does  not  belong 
to  R  ;  let  the  corresponding  elements  of  M  be  j 
and  niQ.  If  r<XQ,  there  is  an  ascending  funda- 
mental series  {r^,,},  which  is  limited  by  x^  and,  from 

a  certain  v^  on, 

r<r^^     for     v^v^. 

The  image  of  {r^J  in  M  is  an  ascending  funda- 
mental series  {^a^}j  which  will  be  limited  by  an  uZq 
of  M,  and  we  have  (§  lo)  s^^  ^  in^  for  every  j/,  and 
•$■  <  ^K  f°^  '^^ ^0-      Thus  (§  7)  J  <  m^. 

U  r>XQ,  we  conclude  similarly  that  s  >-  7/1^. 

Let  us  consider,  finally,  two  elements  Xq  and  x'q 
not  belonging  to  R  and  the  elements  m^  and  m'^ 
corresponding  to  them  in  M  ;  then  we  show,  by 
an  analogous   consideration,    that,    if  Xq  <x'q,    then 

The  proof  of  the  similarity  of  X  and  M  is  now 
finished,  and  we  thus  have 

Halle,  March  1895. 


[207]      CONTRIBUTIONS  TO  THE 
FOUNDING  OF  THE  THEORY  OF 
TRANSFINITE  NUMBERS 

(Second  Article) 

Weil-Ordered  Aggregates 

Among  simply  ordered  aggregates  "well-ordered 
aggregates  "  deserve  a  special  place  ;  their  ordinal 
types,  which  we  call  "ordinal  numbers,"  form  the 
natural  material  for  an  exact  definition  of  the 
higher  transfinite  cardinal  numbers  or  powers, — a 
definition  which  is  throughout  conformable  to  that 
which  was  given  us  for  the  least  transfinite  cardinal 
number  Aleph-zero  by  the  system  of  all  finite 
numbers  i/  (§  6). 

We  call  a  simply  ordered  aggregate  F  (§  7) 
"well-ordered"  if  its  elements /ascend  in  a  definite 
succession  from  a  lowest  f^  in  such  a  way  that  : 

I.  There  is  in  F  an  element /^  which  is  lowest  in 
rank. 

II.  If  F'  is  any  part  of  F  and  if  F  has  one  or 
many  elements  of  higher  rank  than  all  elements 
of  F',  then  there  is  an  element  /'  of  F  which 
follows     immediately    after     the     totality     F',     so 

137 


138     THE  FOU AWING  OF  THE   THEORY 

that  no  elements  in  rank  between  f  and  F'  occur 
in   F.* 

In  particular,  to  every  single  element  /  of  F,  if 
it  is  not  the  highest,  follows  in  rank  as  next  higher 
another  definite  element  f  ;  this  results  from  the 
condition  II  if  for  F'  we  put  the  single  element  / 
Further,  if,  for  example,  an  infinite  series  of  con- 
secutive elements 

e'  <  e"  <  e'"  <  .  .  .  <  e^^^  -<  ^(^+i) .  .  . 

is  contained  in  F  in  such  a  way,  however,  that  there 
are  also  in  F  elements  of  [2o8]  higher  rank  than  all 
elements  e^"^,  then,  by  the  second  condition,  putting 
for  F'  the  totality  {^^"^},  there  must  exist  an  element 
f  sugh  that  not  only 

f  >  e^^^ 

for  all  values  of  v,  but  that  also  there  is  no  element 
g  in  F  which  satisfies  the  two  conditions 

g  >  e"^ 

for  all  values  of  v. 

Thus,  for  example,  the  three  aggregates 
(^1,  a^,  .  .  .,  a^,  .  .  .), 
(«i,  a^,  .  .  .,  a^,  .  .  .,  dj^,  i?2-  •  -y  ^fj^y  •  '  -^ 

where 

*  This  definition  of  "well-ordered  aggregates,"  apart  from  the 
wording,  is  identical  with  that  which  was  introduced  in  vol.  xxi  of  the 
A/at/i.  Attn.,  p.  548  {Grundlas:en  einer  allgcfneinen  Maimichfaltig- 
keitslehre,  p.  4).     [See  Section  VII  of  the  Introduction.] 


OF  TRANSFINITE  NUMBERS        139 

are  well-ordered.  The  two  first  have  no  highest 
element,  the  third  has  the  highest  element  ^3;  in 
the  second  and  third  b^  immediately  follows  all 
the  elements  a^,  in  the  third  i\  immediately  follows 
all  the  elements  a^  and  b'^. 

In  the  following  we  will  extend  the  use  of  the 
signs  -<  and  >^,  explained  in  §  7,  and  there  used 
to  express  the  ordinal  relation  of  two  elements,  to 
groups  of  elements,  so  that  the  formulae 

M-<  N, 
M>N 

are  the  expression  for  the  fact  that  in  a  given  order 
all  the  elements  of  the  aggregate  M  have  a  lower, 
or  higher,  respectively,  rank  than  all  elements  of 
the  aggregate  N. 

A.  Every  part  F^  of  a  well-ordered  aggregate  F 
has  a  lowest  element. 

Proof. — If  the  lowest  element /^  of  F  belongs  to 
Fj,  then  it  is  also  the  lowest  element  of  F^.  In 
the  other  case,  let  F'  be  the  totality  of  all  elements 
of  F""  which  have  a  lower  rank  than  all  elements  F^, 
then,  for  this  reason,  no  element  of  F  lies  between 
F'  and  F^.  Thus,  if/'  follows  (II)  next  after  F, 
then  it  belongs  necessarily  to  I^  and  here  takes  the 
lowest  rank. 

B.  If  a  simply  ordered  aggregate  F  is  such  that 
both  F  and  every  one  of  its  parts  have  a  lowest 
element,  then  F  is  a  well-ordered  aggregate. 

\20()\  Proof. — Since  F  has  a  lowest  element, 
the  condition  I  is  satisfied.      Let  F'  be  a  part  of  F 


140     THE  FOUNDING  OF  THE   THEORY 

such  that  there  are  in  F  one  or  more  elements 
which  follow  F' ;  let  F^  be  the  totality  of  all  these 
elements  and  /'  the  lowest  element  of  F^,  then 
obviously/'  is  the  element  of  F  which  follows  next 
to  F'.  Consequently,  the  condition  II  is  also  satis- 
fied, and  therefore  F  is  a  well-ordered  aggregate. 

C.  Every  part  F'  of  a  well-ordered  aggregate  F 
is  also  a  well-ordered  aggregate. 

Proof. — By  theorem  A,  the  aggregate  F'  as  well 
as  every  part  F''  of  F'  (since  it  is  also  a  part  of  F) 
has  a  lowest  element  ;  thus  by  theorem  B,  the 
aggregate  F'  is  well-ordered. 

D.  Every  aggregate  G  which  is  similar  to  a  well- 
ordered  aggregate  F  is  also  a  well-ordered  aggregate. 

Proof. — If  M  is  an  aggregate  which  has  a  lowest 
element,  then,  as  immediately  follows  from  the 
concept  of  similarity  (§  7),  every  aggregate  N 
similar  to  it  has  a  lowest  element.  Since,  now, 
we  are  to  have  G  rsj  F,  and  F  has,  since  it  is  a 
well-ordered  aggregate,  a  lowest  element,  the  same 
holds  of  G.  Thus  also  every  part  G'  of  G  has  a 
lowest  element  ;  for  in  an  imaging  of  G  on  F,  to 
the  aggregate  G'  corresponds  a  part  F'  of  F  as 
image,   so  that 

G'  00  F'. 

But,  by  theorem  A,  F'  has  a  lowest  element,  and 
therefore  also  G'  has.  Thus,  both  G  and  every 
part  of  G  have  lowest  elements.  By  theorem  B, 
consequently,  G  is  a  well-ordered  aggregate. 

E.  If  in  a  well-ordered  aggregate  G,  in  place  of 


OF  TRANSFINITE  NUMBERS        141 

its  elements  g  well-ordered  aggregates  are  sub- 
stituted in  such  a  way  that,  if  F^^  and  F"^'  are  the 
well-ordered  aggregates  which  occupy  the  places 
of  the  elements  g  and  g'  and  g  -<^  g\  then  also 
F^^  -<  F^^',  then  the  aggregate  H,  arising  by  com- 
bination in  this  manner  of  the  elements  of  all  the 
aggregates  F^,  is  well-ordered. 

Proof. — Both  H  and  every  part  H^  of  H  have 
lowest  elements,  and  by  theorem  B  this  characterizes 
H  as  a  well-ordered  aggregate.  For,  if  g^  is  the 
lowest  element  of  G,  the  lowest  element  of  F^  is 
at  the  same  time  the  lowest  element  of  H.  If, 
further,  we  have  a  part  Hj  of  H,  its  elements 
belong  to  definite  aggregates  F^  which  form,  when 
taken  together,  a  part  of  the  well-ordered  aggre- 
gate {F^},  which  consists  of  the  elements  F^  and 
is  similar  to  the  aggregate  G.  If,  say,  F^  is  the 
lowest  element  of  this  part,  then  the  lowest  element 
of  the  part  of  H^  contained  in  F^  is  at  the  same 
time  the  lowest  element  of  H.  . 


[210]  §   13 

The  Segments  of  Well-Ordered  Aggregates 

If  /  is  any  element  of  the  well-ordered  aggre- 
gate F  which  is  different  from  the  initial  element y^^, 
then  we  will  call  the  aggregate  A  of  all  elements 
of  F  which  precede /"a  "  segment  {AbscJinitt)  of  F, " 
or,  more  fully,  *'  the  segment  of  F  which  is  defined 
by  the  element/"     On  the  other  hand,  the  aggre- 


142     THE  FOUNDING  OF  THE   THEORY 

gate  R  of  all  the  other  elements  of  F,  including  /, 
is  a  ''remainder  of  F,"  and,  more  fully,  ''the 
remainder  which  is  determined  by  the  element  /!" 
The  aggregates  A  and  R  are,  by  theorem  C  of 
§  12,  well-ordered,  and  we  may,  by  §  8  and  §  I2, 
write  : 

(1)  F  =  (A,  R), 

(2)  R  =  (/,  R'), 

(3)  A  <  R. 

R'  is  the  part  of  R  which  follows  the  initial  element 
/  and  reduces  to  o  if  R  has,  besides  /,  no  other 
element. 

For  example,  in  the  well-ordered  aggregate 

the  segment 

and  the  corresponding  remainder 

(<^3,    ^4,    .    .    .    «^  +  2,    .    .    .    ^1,    <^2'    •    •    •  ^H^    '    '    '    ^V    ^2'    ^'3) 

are  determined  by  the  element  a^  ;  the  segment 

(^1,  a^,  .  .  .,  a,,  .  .  .) 
and  the  corresponding  remainder 

{b^,  b^,  .  .  .,  b^,  .  .  .  q,  ^2,  ^3) 

are  determined  by  the  element  b-^  ;  and  the  segment 

(^1,  «2)  •  •  • .  ^^^»  .  .  .  b^,  b^,  ,  .  .,  b^,  .  .  .  q) 


OF  TRANSFINITE  NUMBERS        143 

"remainder 

and  the  corresponding  gogmcnt 

by  the  element  c.^. 

If  A  and  A'  are  two  segments  of  F,/and/'  their 
determining  elements,  and 

(4)  /'  </, 

then  A'  is  a  segment  of  A.      We  call  A'  the  ''  less," 
and  A  the  ''  greater  "  segment  of  F  : 

(5)  A'<A. 

Correspondingly  we  may  say  of  every  A  of  F  that 
it  is  "  less  "  than  F  itself  : 

A<F. 

[211]  A.  If  two  similar  well-ordered  aggregates 
F  and  G  are  imaged  on  one  another,  then  to  every 
segment  A  of  F  corresponds  a  similar  segment  B  of 
G,  and  to  every  segment  B  of  G  corresponds  a 
similar  segment  A  of  F,  and  the  elements /" and  ^ 
of  F  and  G  by  which  the  corresponding  segments 
A  and  B  are  determined  also  correspond  to  one 
another  in  the  imaging. 

Proof. — If  we  have  two  similar  simply  ordered 
aggregates  M  and  N  imaged  on  one  another,  m  and 
n  are  two  corresponding  elements,  and  M'  is  the 
aggregate  of  all  elements  of  M  which  precede  m 
and  N'  is  the  aggregate  of  all  elements  of  N  which 
precede  n^  then  in  the  imaging  M'  and  N'  correspond 
to  one  another.  For,  to  every  element  m'  of  M 
that  precedes  in  must  correspond,  by  §  7,  an  element 


144     THE  FOUNDING  OF  THE   THEORY 

n'  of  N  that  precedes  n^  and  inversely.  If  we  apply 
this  general  theorem  to  the  well-ordered  aggregates 
F  and  G  we  get  what  is  to  be  proved. 

B.  A  well-ordered  aggregate  F  is  not  similar  to 
any  of  its  segments  A. 

Proof. — Let  us  suppose  that  F  oo  A,  then  we  will 
imagine  an  imaging  of  F  on  A  set  up.  By  theorem 
A  the  segment  A'  of  A  corresponds  to  the  segment 
A  of  F,  so  that  A'  oo  A.  Thus  also  we  would  have 
A'  rsj  F  and  A'<  A.  From  A'  would  result,  in  the 
same  manner,  a  smaller  segment  A''  of  F,  such  that 
A''  oo  F  and  A"  <  A' ;  and  so  on.  Thus  we  would 
obtain  an  infinite  series 

A>A'>A".  .  .  A(^)>A('^+i).  .  . 

of  segments  of  F,  which  continually  become  smaller 
and  all  similar  to  the  aggregate  F.  We  will 
denote  by  /,  /',  f'\  .  .  . ,  /^"^  .  .  .  the  elements  of 
F  which  determine  these  segments  ;  then  we  would 
have 

/>/'  >/"  >■■■  >/<-*  >  A+i) . . . 

We  would  therefore  have  an  infinite  part 

of  F  in  which  no  element  takes  the  lowest  rank. 
But  by  theorem  A  of  §  12  such  parts  of  F  are  not 
possible.  Thus  the  supposition  of  an  imaging  F  on 
one  of  its  segments  leads  to  a  contradiction,  and 
consequently  the  aggregate  F  is  not  similar  to  any 
of  its  segments. 


OF  TRANSFINITE  NUMBERS        145 

Though  by  theorem  B  a  well-ordered  aggregate 
F  is  not  similar  to  any  of  its  segments,  yet,  if  F  is 
infinite,  there  are  always  [212]  other  parts  of  F  to 
which  F  is  similar.     Thus,  for  example,  the  aggregate 

F  =  (^i,  a^,  .  .  .,  «,,,  .  .  .) 

is  similar  to  every  one  of  its  remainders 

Consequently,  it  is  important  that  we  can  put  by  the 
side  of  theorem  B  the  following  : 

C.  A  well-ordered  aggregate  F  is  similar  to  no 
part  of  any  one  of  its  segments  A. 

Proof. — Let  us  suppose  that  F'  is  a  part  of  a 
segment  A  of  F  and  F'  00  F.  We  imagine  an 
imaging  of  F  on  F' ;  then,  by  theorem  A,  to  a 
segment  A  of  the  well-ordered  aggregate  F  corre- 
sponds as  image  the  segment  Y"  of  F'  ;  let  this 
segment  be  determined  by  the  element  f  of  F'. 
The  element  /'  is  also  an  element  of  A,  and  de- 
termines a  segment  A'  of  A  of  which  F'"  is  a  part. 
The  supposition  of  a  part  F'  of  a  segment  A  of  F 
such  that  F'  00  F  leads  us  consequently  to  a  part  F" 
of  a  segment  A'  of  A  such  that  Y"  00  A.  The  same 
manner  of  conclusion  gives  us  a  part  Y'"  of  a 
segment  A"  of  A'  such  that  F"'  00  A'.  Proceeding 
thus,  we  get,  as  in  the  proof  of  theorem  B,  an 
infinite  series  of  segments  of  F  which  continually 
become  smaller  : 

A>A'>A".  .  .  A(''>>A<''+i>.  .  ., 

10 


146     THE  FOUNDING  OF  THE   THEORY 

and  thus  an  infinite  series  of  elements  determining 
these  segments  : 

in  which  is  no  lowest  element,  and  this  is  impossible 
by  theorem  A  of  §  12.  Thus  there  is  no  part  F' 
of  a  segment  A  of  F  such  that  F'  00  F. 

D.  Two  different  segments  A  and  A'  of  a  well- 
ordered  aggregate  F  are  not  similar  to  one  another. 

Proof. — If  A'<A,  then  A'  is  a  segment  of  the 
well-ordered  aggregate  A,  and  thus,  by  theorem  B, 
cannot  be  similar  to  A. 

E.  Two  similar  well-ordered  aggregates  F  and  G 
can  be  imaged  on  one  another  only  in  a  single 
manner. 

Proof. — Let  us  suppose  that  there  are  two  different 
imagings  of  F  on  G,  and  let /be  an  element  of  F  to 
which  in  the  two  imagings  different  images  g  and  g 
in  G  correspond.  Let  A  be  the  segment  of  F  that 
is  determined  by/,  and  B  and  B'  the  segments  of  G 
that  are  determined  by  g  and  g .  By  theorem  A, 
both  A  00  B  [213]  and  A  00  B',  and  consequently 
BooB',  contrary  to  theorem  D. 

F.  If  F  and  G  are  two  well-ordered  aggregates, 
a  segment  A  of  F  can  have  at  most  one  segment 
B  in  G.  which  is  similar  to  it. 

Proof — If  the  segment  A  of  F  could  have  two 
segments  B  and  B'  in  G  which  were  similar  to  it,  B 
and  B'  would  be  similar  to  one  another,  which  is 
impossible  by  theorem  D. 

G.  If  A  and  B  are  similar  segments  of  two  well- 


OP  TRANSFJNITR  NUMBERS        147 

ordered  aggregates  F  and  G,  for  every  smaller 
segment  A'<A  of  F  there  is  a  similar  segment 
B'  <  ]^  of  G  and  for  every  smaller  segment  W  <  B  of 
G  a  similar  segment  A'  <  A  of  F. 

The  proof  follows  from  theorem  A  applied  to  the 
similar  aggregates  A  and  B. 

H.  If  A  and  A'  are  two  segments  of  a  well- 
ordered  aggregate  F,  B  and  B'  are  two  segments 
similar  to  those  of  a  well-ordered  aggregate  G,  and 
A'<A,  then  B'<B. 

The  proof  follows  from  the  theorems  F  and  G. 

I.  If  a  segment  B  of  a  well-ordered  aggregate  G 
is  similar  to  no  segment  of  a  well-ordered  aggregate 
F,  then  both  every  segment  B'  >  B  of  D  and  G  itself 
are  similar  neither  to  a  segment  of  F  nor  F  itself. 

The  proof  follows  from  theorem  G. 

K.  If  for  any  segment  A  of  a  well-ordered 
aerereeate  F  there  is  a  similar  see^ment  B  of  another 
well-ordered  aggregate  G,  and  also  inversely,  for 
every  segment  B  of  G  a  similar  segment  A  of  F, 
then  F  00  G. 

Proof. — We  can  image  F  and  G  on  one  another 
according  to  the  following  law  :  Let  the  lowest 
element  f^  of  F  correspond  to  the  lowest  element  g^ 
of  G.  If  f^fi  is  any  other  element  of  F,  it 
determines  a  segment  A  of  F.  To  this  segment 
belongs  by  supposition  a  definite  similar  segment 
B  of  G,  and  let  the  element  ^  of  G  which  determines 
the  segment  B  be  the  image  of  F.  And  if  g  is  any 
element  of  G  that  follows  g^,  it  determines  a 
segment  B  of  G,  to  which  by  supposition  a  similar 


148     THE  FOUNDING  OF  THE   THEORY 

segment  A  of  F  belongs.  Let  the  element /"which 
determines  this  segment  A  be  the  image  of  ^.  It 
easily  follows  that  the  bi-univocal  correspondence  of 
F  and  G  defined  in  this  manner  is  an  imaging  in  the 
sense  of  §  7.  For  if/  and/'  are  any  two  elements 
of  F,  g  and  g'  [2 1 4]  the  corresponding  elements  of 
G,  A  and  A'  the  segments  determined  by/  and  /', 
B  and  B'  those  determined  by  g  and  g\  and  if,  say, 

/'</- 

then 

A'<A. 

By  theorem  H,  then,  we  have 

B'<B, 

and  consequently 

L.  If  for  every  segment  A  of  a  well-ordered 
aggregate  F  there  is  a  similar  segment  B  of  another 
well-ordered  aggregate  G,  but  if,  on  the  other  hand, 
there  is  at  least  one  segment  of  G  for  which  there  is 
no  similar  segment  of  F,  then  there  exists  a  definite 
segment  B^  of  G  such  that  B^OoF. 

Proof. — Consider  the  totality  of  segments  of  G  for 
which  there  are  no  similar  segments  in  F.  Amongst 
them  there  must  be  a  least  segment  which  we  will  call 
B^.  This  follows  from  the  fact  that,  by  theorem  A 
of  §  12,  the  aggregate  of  all  the  elements  determin- 
ing these  segments  has  a  lowest  element  ;  the 
segment  B^  of  G  determined  by  that  element  is  the 
least  of  that  totality.     By  theorem  I,  every  segment 


OF  JRANSFINITE  NUMBERS         149 

of  G  which  is  greater  than  B^  is  such  that  no  segment 
similar  to  it  is  present  in  F.  Thus  the  segments 
B  of  G  which  correspond  to  similar  segments  of  F 
must  all  be  less  than  Bj,  and  to  every  segment 
B  <  B^  belongs  a  similar  segment  A  of  F,  because 
Bj  is  the  least  segment  of  G  among  those  to  which 
no  similar  segments  in  F  correspond.  Thus,  for 
every  segment  A  of  F  there  is  a  similar  segment  B  of 
B^,  and  for  every  segment  B  of  B^  there  is  a  similar 
segment  A  of  F.      By  theorem  K,  we  thus  have 

FooB^. 

M.  If  the  well-ordered  aggregate  G  has  at  least 
one  segment  for  which  there  is  no  similar  segment 
in  the  well-ordered  aggregate  F,  then  every  segment 
A  of  F  must  have  a  segment  B  similar  to  it  in  G. 

Proof. — Let  B^  be  the  least  of  all  those  segments 
of  G  for  which  there  are  no  similar  segments  in  F.  * 
If  there  were  segments  in  F  for  which  there  were  no 
corresponding  segments  in  G,  amongst  these,  one, 
which  we  will  call  A^,  would  be  the  least.  For 
every  segment  of  Aj^  would  then  exist  a  similar 
segment  of  B^^,  and  also  for  every  segment  of  B^^  a 
similar  segment  of  A^.  Thus,  by  theorem  K,  we 
would  have 

B^  no  A^. 

[215]  But  this  contradicts  the  datum  that  for  B^ 
there  is  no  similar  segment  of  F.  Consequently, 
there  cannot  be  in  F  a  segment  to  which  a  similar 
segment  in  G  does  not  correspond. 

*  See  the  above  proof  of  L. 


ISO     THE  FOUNDING  OF  THE   THEORY 

N.  If  F  and  G  are  any  two  well-ordered  aggre- 
gates, then  either  : 

{a)  F  and  G  are  similar  to  one  another,  or 

{b)  there  is  a  definite  segment  Bj  of  G  to  which 
F  is  similar,  or 

(<:)  there  is  a  definite  segment  A^  of  F  to  which 
G  is  similar  ; 
and  each  of  these  three  cases  excludes  the  two  others. 

Proof. — The  relation  of  F  to  G  can  be  any  one  of 
the  three  : 

(a)  To  every  segment  A  of  F  there  belongs  a 
similar  segment  B  of  G,  and  inversely,  to  every 
segment  B  of  G  belongs  a  similar  one  A  of  F  ; 

{b)  To  every  segment  A  of  F  belongs  a  similar 
segment  B  of  G,  but  there  is  at  least  one  segment 
of  G  to  which  no  similar  segment  in  F  corresponds ; 

{c)  To  every  segment  B  of  G  belongs  a  similar 
segment  A  of  F,  but  there  is  at  least  one  segment  of 
F  to  which  no  similar  segment  in  G  corresponds. 

The  case  that  there  is  both  a  segment  of  F  to 
which  no  similar  segment  in  G  corresponds  and  a 
segment  of  G  to  which  no  similar  segment  in  F 
corresponds  is  not  possible  ;  it  is  excluded  by 
theorem  M. 

By  theorem  K,  in  the  first  case  we  have 

F  cNj  G. 

In  the  second  case  there  is,  by  theorem  L,  a  definite 
segment  V>^  of  B  such  that 

BiOoF; 


OF  TRANSFINITE  NUMBERS        151 

and  in  the  third  case  there  is  a  definite  segment  A^ 
of  F  such  that 

Ai  fxj  G. 

We  cannot  have  F  c\j  G  and  F  00  B^  simultaneously, 
for  then  we  would  have  G  00  Bj,  contrary  to  theorem 
B  ;  and,  for  the  same  reason,  we  cannot  have  both 
F  rv)  G  and  G  00  A^.  Also  it  is  impossible  that 
both  F  00  Bj  and  G  00  A^,  for,  by  theorem  A, 
from  F  00  B^  would  follow  the  existence  of  a 
segment  B'^  of  B^^  such  that  A^  00  B\.  Thus  we 
would  have  G  00  B'j,  contrary  to  theorem  B. 

O.  If  a  part  F'  of  a  well-ordered  aggregate  F  is 
not  similar  to  any  segment  of  F,  it  is  similar  to  F 
itself. 

Proof. — By  theorem  C  of  §  12,  F'  is  a  well-ordered 
aggregate.  If  F'  were  similar  neither  to  a  segment 
of  F  nor  to  F  itself,  there  would  be,  by  theorem  N, 
a  segment  F'^  of  F'  which  is  similar  to  F.  But  F'^ 
is  a  part  of  that  segment  A  of  F  which  [216]  is 
determined  by  the  same  element  as  the  segment  F'^ 
of  F'.  Thus  the  aggregate  F  would  have  to  be 
similar  to  a  part  of  one  of  its  segments,  and  this 
contradicts  the  theorem  C. 

§  14 

The  Ordinal  Numbers  of  Weil-Ordered 

Aggregates 

By  §  7,  every  simply  ordered  aggregate  M  has  a 
definite  ordinal  type  M  ;  this  type  is  the  general  con- 


152     THE  FOUNDING  OF  THE   THEORY 

cept  which  results  from  M  if  we  abstract  from  the 
nature  of  its  elements  while  retaining  their  order  of 
precedence,  so  that  out  of  them  proceed  units 
{Einsefi)  which  stand  in  a  definite  relation  of  pre- 
cedence to  one  another.  All  aggregates  which  are 
similar  to  one  another,  and  only  such,  have  one  and 
the  same  ordinal  type.  We  call  the  ordinal  type  of 
a  well-ordered  aggregate  F  its  "ordinal  number." 

If  a  and  /3  are  any  two  ordinal  numbers,  one  can 
stand  to  the  other  in  one  of  three*  possible  relations. 
For  if  F  and  G  are  two  well-ordered  aggregates 
such  that 

F  =  a,     G  =  /3, 

then,  by  theorem  N  of  §  13,  three  mutually  ex- 
clusive cases  are  possible  : 

{a)  F  00  G ; 

{b)  There  is  a  definite  segment  Bj  of  G  such  that 

FooB^; 
(c)  There  is  a  definite  segment  Aj  of  F  such  that 

Goo  A^. 

As  we  easily  see,  each  of  these  cases  still  subsists 
if  F  and  G  are  replaced  by  aggregates  respectively 
similar  to  them.  Accordingly,  we  have  to  do  with 
three  mutually  exclusive  relations  of  the  types  a 
and  )8  to  one  another.  In  the  first  case  a  =  /3;  in 
the  second  we  say  that  a</5;  in  the  third  we  say 
that  a>/3.     Thus  we  have  the  theorem  : 


OF  TRANSFINITE  NUMBERS         153 

A.  If  a  and  ^  are  any  two  ordinal  numbers,  we 
have  either  a  =  l3  or  a<^  or  a> /3. 

From  the  definition  of  minority  and  majority 
follows  easily  : 

B.  If  we  have  three  ordinal  numbers  a,  /5,  y,  and 
if  a  <  18  and  ^  <y,  then  a  <  y. 

Thus  the  ordinal  numbers  form,  when  arranged 
in  order  of  magnitude,  a  simply  ordered  aggregate  ; 
it  will  appear  later  that  it  is  a  well-ordered  aggre- 
gate. 

[217]  The  operations  of  addition  and  multipli- 
cation of  the  ordinal  types  of  any  simply  ordered 
aggregates,  defined  in  §  8,  are,  of  course,  applicable 
to  the  ordinal  numbers.  If  a  =  F  and  /3  =  G,  where 
F  and  G  are  two  well-ordered  aggregates,  then 


(1)  a  +  /3  =  (F,  G). 

The  aggregate  of  union  (F,  G)  is  obviously  a 
well-ordered  aggregate  too  ;  thus  we  have  the 
theorem  : 

C.  The  sum  of  two  ordinal  numbers  is  also  an 
ordinal  number. 

In  the  sum  a  +  l3,  a  is  called  the  "augend"  and 
^  the  ''addend." 

Since  F  is  a  segment  of  (F,  G),  we  have  always 

(2)  •     a<a  +  /3. 

On  the  other  hand,  G  is  not  a  segment  but  a  re- 
mainder of  (F,  G),  and  may  thus,  as  we  saw  in 
§   13,  be  similar  to  the  aggregate  (F,  G).      If  this 


154     THE  FOUNDING  OF  THE   THEORY 

is  not  the  case,  G  is,  by  theorem  O  of  §  13,  similar 
to  a  segment  of  (F,  G).      Thus 

(3)  ^<«+/3. 

Consequently  we  have  : 

D.  The  sum  of  the  two  ordinal  numbers  is  always 
greater  than  the  augend,  but  greater  than  or  equal 
to  the  addend.  If  we  have  a  +  ^  =  a  +  y,  we  always 
have  ,8  =  y. 

In  general  a-\-^  and  /3  +  a  are  not  equal.  On 
the  other  hand,  we  have,  if  y  is  a  third  ordinal 
number, 

(4)  (a  +  ^)  +  y  =  a  +  (/3  +  y). 

That  is  to  say  : 

E.  In  the  addition  of  ordinal  numbers  the  associa- 
tive law  always  holds. 

If  we  substitute  for  every  element  g  of  the 
aggregate  G  of  type  /3  an  aggregate  F^^  of  type  a, 
we  get,  by  theorem  E  of  §  12,  a  well-ordered 
aggregate  H  whose  type  is  completely  determined 
by  the  types  a  and  /3  and  will  be  called  the  product 

(5)  "  i^^^=«^ 

(6)  a.^  =  H. 

F.  The  product  of  two  ordinal  numbers  is  also 
an  ordinal  number. 

In  the  product  a  .^8,  a  is  called  the  ''  multiplicand  " 
and  /3  the  "multiplier." 

In  general  a./^and  /S.aare  not  equal.  But  we 
have  (§  8) 


OF  TRANS  FINITE  NUMBERS         155 

(7)  (a.^).y  =  a.(/3.y). 

That  is  to  say  : 

[218]   G.  In  the  multipHcation  of  ordinal  numbers 

the  associative  law  holds. 

The  distributive  law  is  valid,  in  general  (§  8), 
only  in  the  following  form  : 

(8)  a.(/3  +  y)  =  a.^  +  a.y. 

With  reference  to  the  magnitude  of  the  product, 
the  following  theorem,  as  we  easily  see,  holds  : 

H.  If  the  multipHer  is  greater  than  i,  the  product 
of  two  ordinal  numbers  is  always  greater  than 
the  multiplicand,  but  greater  than  or  equal  to  the 
multiplier.  If  we  have  a.^  =  a.y,  then  it  always 
follows  that  /8  =  y. 

On  the  other  hand,  we  evidently  have 

(9)  a  .  I  =  I  .  a  =  a. 

We  have  now  to  consider  the  operation  of  sub- 
traction. If  a  and  /3  are  two  ordinal  numbers,  and 
a  is  less  than  /5,  there  always  exists  a  definite 
ordinal  number  which  we  will  call  /3  —  a,  which 
satisfies  the  equation 

(10)  a  +  (/3-a)  =  ^. 

For  if  G  =  ^,  G  has  a  segment  B  such  that  B  =  a  ; 
we  call  the  corresponding  remainder  S,  and  have 

G  =  (B,  S), 

/3  =  a  +  S; 


156     THE  FOUNDING  OF  THE   THEORY 

and  therefore 

(ii)  /5-a  =  S. 

The  determinateness  of  ^-a  appears  clearly  from 
the  fact  that  the  segment  B  of  G  is  a  completely 
definite  one  (theorem  D  of  §  13),  and  consequently 
also  S  is  uniquely  given. 

We  emphasize  the  following  formulae,  which 
follow  from  (4),  (8),  and  (lo)  : 

(12)  (7  +  /3)-(y  +  a)  =  ^-a, 

(13)  y(/3-a)  =  y^-ya. 

It  is  important  to  reflect  that  an  infinity  of 
ordinal  numbers  can  be  summed  so  that  their  sum 
is  a  definite  ordinal  number  which  depends  on  the 
sequence  of  the  summands.      If 

is  any  simply  infinite  series  of  ordinal  numbers,  and 
we  have 

(14)  ^v=^v^ 

[219]  then,  by  theorem  E  of  §  12, 

(15)  .G  =  (Gi,    G2,  .  .  .,    G^,   .  .  .) 

is  also  a  well-ordered  aggregate  whose  ordinal 
number  represents  the  sum  of  the  numbers  /3,,. 
We  have,  then, 

(16)  A  +  ^2+  •  •  •  +A.+  .  .  .  =G  =  /3, 

and,  as  we  easily  see  from  the  definition  of  a 
product,  we  always  have 


OF  TRANSFINITE  NUMBERS         157 

(17)  y.(/5i  +  ^,+  ...  +/'i+...) 

If  we  put 

(18)  «^^^^  +  /3^  +  .  .  .   +/^^, 
then 


(19.)  a,  =  (Gi,  G2,  .  .  .  G,). 

We  have 

(20)  a.+i>a., 

and,  by  (10),  we  can  express  the  numbers  P^,  by 
the  numbers  a^  as  follows  : 

(21)  i^i=ai;    /5^+i  =  a^+i  — a^. 

The  series 

«!,  aa,  .  .  .,  a^,  .  .  . 

thus  represents  any  infinite  series  of  ordinal  numbers 
which  satisfy  the  condition  (20)  ;  we  will  call  it  a 
"fundamental  series"  of  ordinal  numbers  (§10). 
Betw^een  it  and  ^  subsists  a  relation  which  can  be 
expressed  m  the  following  manner  : 

{a)  The  number  /5  is  greater  than  a,  for  every 
J/,  because  the  aggregate  (G^,  Gg,  .  .  .,  G„),  whose 
ordinal  number  is  a^,  is  a  segment  of  the  aggregate 
G  which  has  the  ordinal  number  ^  ; 

{b)  If  /3'  is  any  ordinal  number  less  than  ^,  then, 
from  a  certain  v  onwards,  we  always  have 

For,    since    S'  <  jS,    there    is    a    segment    B'    of    the 


158     THE  FOVNDING  OF  THE   THEORY 

aggregate  G  which  is  of  type  /3'.  The  element  of 
G  which  determines  this  segment  must  belong  to 
one  of  the  parts  G^ ;  we  will  call  this  part  G^,^.  But 
then  B'  is  also  a  segment  of  (G^,  Gg,  .  .  .,  G^ ),  and 
consequently  /3'  <  a^, .      Thus 

for  v^v^. 

Thus  /3  is  the  ordinal  number  which  follows  next 
in  order  of  magnitude  after  all  the  numbers  a^  ; 
accordingly  we  will  call  it  the  "limit"  {Grenze)  of 
the  numbers  a^  for  increasing  v  and  denote  it  by 
Lim  a^,  so  that,  by  (i6)  and  (21)  : 

(22)  Lim  a^  =  ai  +  («2  -  «!>  +  •"  •  •  +  (a^+i  -«.)  +  •  •  . 

[220]  We  may  express  what  precedes  in  the 
following  theorem  : 

1.  To  every  fundamental  series  {a,J  of  ordinal 
numbers  belongs  an  ordinal  number  Lim  a,,  which 

V 

follows  next,  in  order  of  magnitude,  after  all  the 
numbers  a„ ;  it  is  represented  by  the  formula  (22). 

If  by  y  we  understand  any  constant  ordinal 
number,  we  easily  prove,  by  the  aid  of  the  formulae 
(12),  (13),  and  (17),  the  theorems  contained  in  the 
formulae  : 

(23)  Lim  (y  +  a„)  =  y  +  Lim  a,  ; 

V  V 

(24)  Lim  y  .  ay  =  y  .  Lhn  a^. 

V  V 

We  have  already  mentioned  in  §  7  that  all  simply 


OF  TRANSFINITE  NUMBERS        159 

ordered  aggregates  of  given  finite  cardinal  number 
V  have  one  and  the  same  ordinal  type.  This  may 
be  proved  here  as  follows.  Every  simply  ordered 
aggregate  of  finite  cardinal  number  is  a  well-ordered 
aggregate  ;  for  it,  and  every  one  of  its  parts,  must 
have  a  lowest  element, — and  this,  by  theorem  B 
of  §  12,  characterizes  it  as  a  well-ordered  aggregate. 
The  types  of  finite  simply  ordered  aggregates  are 
thus  none  other  than  finite  ordinal  numbers.  But 
two  different  ordinal  numbers  a  and  ^  cannot  belong 
to  the  same  finite  cardinal  number  v.  For  if,  say, 
a</3  and  G  =  /3,  then,  as  we  know,  there  exists  a 
segment  B  of  G  such  that  B  =  a.  Thus  the  aggre- 
gate G  and  its  part  B  would  have  the  same  finite 
cardinal  number  v.  But  this,  by  theorem  C  of  §  6, 
is  impossible.  Thus  the  finite  ordinal  numbers 
coincide  in  their  properties  with  the  finite  cardinal 
numbers. 

The  case  is  quite  different  with  the  transfinite 
ordinal  numbers  ;  to  one  and  the  same  transfinite 
cardinal  number  a  belong  an  infinity  of  ordinal 
numbers  which  form  a  unitary  and  connected 
system.  We  will  call  this  system  the  "number- 
class  Z(a),"  and  it  is  a  part  of  the  class  of  types 
[a]  of  §  7.  The  next  object  of  our  consideration  is 
the  number-class  Z(h?Q),  which  we  will  call  "the 
second  number-class."  For  in  this  connexion  we 
understand  by  "the  first  number-class"  the  totality 
[v]  of  finite  ordinal  numbers. 


i6o     THE  FOUNDING  OF  THE  THEORY 
[221]  %   15 

The  Numbers  of  the  Second  Number- Class  Z({^o) 

The  second  number-class  Z(}^q)  is  the  totality  {a} 
of  ordinal  types  a  of  well-ordered  aggregates  of 
the  cardinal  number  «o  (§  6). 

A.  The  second  number-class  has  a  least  number 
0)  =  Lim  V. 

V 

Pi'oof. — By  o)  we  understand  the  type  of  the 
well-ordered  aggregate 

(1)  Fo  =  (/i,  /„  .  .  .,  /,,  .  .  .), 

where  v  runs  through  all  finite  ordinal  numbers  and 

(2)  /.-</.+i^ 
Therefore  (§  7) 

(3)  w=Fo> 

and  (§  6) 

(4)  ^  =  f^o- 

Thus  0)  is  a  number  of  the  second  number-class, 
and  indeed  the  least.  For  if  y  is  any  ordinal 
number  less  than  o),  it  must  (§  14)  be  the  type  of 
a  segment  of  Fq.      But  F^  has  only  segments 

A  =  (/i,/2,  .  .  .,/.), 
with  finite  ordinal  number  v.      Thus  y  =  v.      There- 
fore there  are  no  transfinite  ordinal  numbers  which 
are  less  than  w,  and  thus  w  is  the  least  of  them. 
By    the    definition    of    Lim    a^    given    in    §  14,    we 

V 

obviously  have  a)=Lim  v. 


OF  TRANSFINITE  NUMBERS        i6i 

B.  If  a  is  any  number  of  the  second  number-class, 
the  number  a+i  follows  it  as  the  next  greater 
number  of  the  same  number-class. 

Proof. — Let  F  be  a  well-ordered  aggregate  of 
the  type  a  and  of  the  cardinal  number  Nq  : 

(5)  F  =  a, 

(6)  a=No- 

We  have,  where  by  g  is  understood  a  new  element, 


(7)  a+i=(F,  ^). 
Since  F  is  a  segment  of  (F,  g),  we  have 

(8)  a+i>a. 
We  also  have 


a+i=a+i=No+i=«o  (§^)- 

Therefore  the  number  a+i  belongs  to  the  second 
number-class.  Between  a  and  a+i  there  are  no 
ordinal  numbers  ;  for  every  number  y  [222]  which 
is  less  than  a+  i  corresponds,  as  type,  to  a  segment 
of  (F,  g),  and  such  a  segment  can  only  be  either 
F  or  a  segment  of  F.  Therefore  y  is  either  equal 
to  or  less  than  a. 

C.  If  ai,  aa,  .  .  .,  a„  .  .  .  is  any  fundamental  series 
of  numbers  of  the  first  or  second  number-class,  then 
the  number  Lim  a,  (§  H)  following  them  next  in 

V 

order  of  magnitude  belongs  to  the  second  number- 
class. 

Proof. — By   §  14   there    results   from   the   funda- 

II 


1 62     THE  FOUNDING  OF  THE   THEORY 
mental  series  {a,,}  the  number  Lim  a^  if  we  set  up 

V 

another  series  ^j,  /^g,  •  .  .,  /8^,  .  .  .,  where 

If,  then,  Gj,  Gg,  .  .  .,  G^,  .  .  .  are  well-ordered  aggre- 
gates such  that 

then  also 

G  =  (^i)  ^2,  .  .  .,  G^,  .  .  .) 

is  a  well-ordered  aggregate  and 
Lim  a^  =  G. 

V 

It  only  remains  to  prove  that 

Since  the  numbers  /^j,   ^82,  .  .  .,  ft,  •  •  .    belong   to 
the  first  or  second  number-class,  we  have 

and  thus 

G<h?o  •  No  =  ^^o- 
But,  in  any  case,  G  is  a  transfinite  aggregate,  and 
so  the  case  G<^?q  is  excluded. 

We  will  call  two  fundamental  series  {a^}  and  {a^\ 
of  numbers  of  the  first  or  second  number-class  (§  lO) 
"coherent,"  in  signs  : 

(9)  {«.}  11  {«'.}. 

if  for  every  v  there  are   finite   numbers  Xq   and  ^^ 
such  that 

(10)  aK>OLy,      X>Xo> 


OF  rRANSFINTTE  NUMBERS         163 

and 

(11)  a^>a\,      At^Mo- 

[223]  D.  The  limiting  numbers  Lim  a^  and  Lim  a\ 

V  V 

belonging  respectively  to  two  fundamental  series 
{ay}  and  [a\}  are  equal  when,  and  only  when, 
{a.}  II  {«;}. 

Proof.  —  For  the  sake  of  shortness  we  put 
Lim   a^,  =  /3,    Lim   a'^  =  y.        We    will    first    suppose 

that  {a^  II  {a'4  ;  then  we  assert  that  /3  =  y.  For 
if  /3  were  not  equal  to  y,  one  of  these  two  numbers 
would  have  to  be  the  smaller.  Suppose  that  /8<y. 
From  a  certain  v  onwards  we  would  have  a'v>/3 
(§  14),  and  consequently,  by  (11),  from  a  certain 
/x  onwards  we  would  have  a^>/5.  But  this  is 
impossible    because   /3=Lim  a^.      Thus   for   all   ^'s 

V 

we  have  a^</3. 

If,  inversely,  we  suppose  that  /3  =  y,  then,  because 
oiv<y^  we  must  conclude  that,  from  a  certain  X 
onwards,  a\>a^,  and,  because  a\<l3,  we  must 
conclude  that,  from  a  certain  ^  onwards,  afj^>  a\. 
That  is  to  say,  {aj  ||  {a\]. 

E.  If  a  is  any  number  of  the  second  number- 
class  and  vq  any  finite  ordinal  number,  we  have 
j/^-|-a  =  a,  and  consequently  also  a  —  VQ  =  a. 

Proof. — We  will  first  of  all  convince  ourselves  of 
the  correctness  of  the  theorem  when  a  =  no.  We 
have 


^0  =  Uv  <^2'    •    •    •  <r  J' 


1 64     THE  FOUNDING  OF  THE  THEORY 
and  consequently 

But  if  a>a),  we  have 

a  =  CO  +  (a  —  w), 

i/Q  +  a  =  (I'o  +  w)  +  («  ""  w)  =  ^  +  («  ~  ^^  =  «• 
F.    If  j/Q   is   any  finite   ordinal    number,  we   have 

I/q     .      ft)  =   ftj. 

Ptoof, — In  order  to  obtain  an  aggregate  of  the 
type  i/Q .  o)  we  have  to  substitute  for  the  single 
elements  /,  of  the  aggregate  (/i,  Z^,  ...,/„...) 
aggregates  {g,^  i,  ^v,  2>  •  •  •  >  gv,  v)  of  the  type  v^.  We 
thus  obtain  the  aggregate 

(^1. 1'  ^1, 2)  •  •  -5  S\,  V  ^2.  i»  •  •  • '  ^2,  "o'  •  •  • '  «^*'.  1' 

which  is    obviously  similar  to  the  aggregate  {/,}. 
Consequently 

I/q  .  0)  =  ft). 

The  same  result  is  obtained  more  shortly  as  follows. 
By  (24)  of  §  14  we  have,  since  ft)=Lim  j/, 

1/q  0)=  Lim  j^Q  I/. 
On  the  other  hand, 

and  consequently 

Lim  j/Qy=Lim  t/  =  ft)  ; 
1/  I' 

so  that 


OF  TRANSFINITE  NUMBERS        165 

[224]   G.    We  have  always 

(a  +  ^0)0)  =  aw, 


where    a  is  a  number  of   the  second  number-class 
d  i/q  a  number  of  1 
Proof. — We  have 


and  Vq  a  number  of  the  first  number-class. 


Lim  i/  =  ft). 

V 

By  (24)  of  §  14  we  have,  consequently, 

(a  +  i/o)a)  =  Lim  {a-\-v^v. 

But 

I  2  V 

(a4-i/o>=        (a  +  i/o)  +  («  +  »'o)+  •  •  •  +(«  +  t^o) 
I  2  v—l 

=  a  +  (^^o  +  «)  +  K  +  «)  •  •  •  (t^o  +  «)  +  ^'o 
I       2  1/ 

=  a  +  a+...+a  +  i/o 

=  aj/  +  i/q. 

Now  we  have,  as  is  easy  to  see, 

{av\-VQ}  II  {av}, 
and  consequently 

Lim  (a  +  i^o)i/=  Lim  (ai^  +  i/o)  =  Lim  a»/  =  aaj. 

V  V  V 

H.  If  a  is  any  number  of  the  second  number- 
class,  then  the  totality  {a]  of  numbers  a  of  the 
first  and  second  number-classes  which  are  less  than 
a  form,  in  their  order  of  magnitude,  a  well-ordered 
aggregate  of  type  a. 


1 66     THE  FOUNDING  OF  THE   THEORY 

Proof. — Let  F  be  a  well-ordered  aggregate  such 
that  F  =  a,  and  let/^  be  the  lowest  element  of  F.  If 
a  is  any  ordinal  number  which  is  less  than  a,  then, 
by  §  14,  there  is  a  definite  segment  A'  of  F  such 
that  _ 

A'  =  a, 

and  inversely  every  segment  A'  determines  by  its 
type  h!  =  0!  a  number  a  <a  of  the  first  or  second 

number-class.  For,  since  F  =  ^?Q,  A'  can  only  be 
either  a  finite  cardinal  number  or  ^q.  The  segment 
A'  is  determined  by  an  element/' >^/i  of  F,  and 
inversely  every  element/'  >-/i  of  F  determines  a 
segment  A'  of  F.  If/'  and/"  are  two  elements  of 
F  which  follow  /  in  rank,  A'  and  A"  are  the 
segments  of  F  determined  by  them,  a  and  a'  are 
their  ordinal  types,  and,  say/'  -</",  then,  by  §  13, 
A'<A"  and  consequently  a' <  a".  [225]  If,  then, 
we  put  F  =  (/,  F'),  we  obtain,  when  we  make  the 
element/'  of  F'  correspond  to  the  element  a  of  {a'}, 
an  imaging  of  these  two  aggregates.      Thus  we  have 

R}^F'. 

ButF'  =  a— I,  and,  by  theorem  E,  a— i=a.  Con- 
sequently 

{a)=a. 

Since  a  =  «o,  we  also  have  {a'}  =  t^o;  thus  we  have 
the  theorems  : 

I.    The    aggregate    {a'}    of    numbers    a     of    the 
first  and  second  number-classes  which  are  smaller 


OF  TRANSFINITE  NUMBERS         167 

than  a  number  a  of  the    second   number-class  has 
the  cardinal  number  j^^. 

K.  Every  number  a  of  the  second  number-class 
is  either  such  that  {a)  it  arises  out  of  the  next 
smaller  number  a.^  by  the  addition  of  i  : 


or  {ii)  there  is  a  fundamental  series  {a^  of  numbers 
of  the  first  or  second  number-class  such  that 

a  —  Lim  a^. 

V 

Proof. — Let  a  =  F.  If  F  has  an  element  g  which 
is  highest  in  rank,  we  have  F  =  (A,  g),  where  A  is 
the  segment  of  F  which  is  determined  by  g.  We 
have  then  the  first  case,  namely, 

a  =  A-f  i=a_iH-  I. 

There   exists,    therefore,    a    next    smaller    number 
which  is  that  called  %. 

But  if  F  has  no  highest  element,  consider  the 
totality  [a]  of  numbers  of  the  first  and  second 
number-classes  which  are  smaller  than  a.  By 
theorem  H,  the  aggregate  {a'},  arranged  in  order  of 
magnitude,  is  similar  to  the  aggregate  F  ;  among 
the  numbers  d,  consequently,  none  is  greatest.  By 
theorem  I,  the  aggregate  {a\  can  be  brought  into 
the  form  {ay\  of  a  simply  infinite  series.  If  we  set 
out  from  a'l,  the  next  following  elements  ai^  a'3,  .  .  . 
in  this  order,  which  is  different  from  the  order  of 
magnitude,  will,  in  general,  be  smaller  than  a^  ; 
but    in    every   case,    in   the   further   course    of   the 


1 68     THE  FOUNDING  OF  THE   THEORY 

process,  terms  will  occur  which  are  greater  than  a'^  ; 

for   a'l    cannot    be    greater    than    all    other    terms, 

because    among    the    numbers    {a'^}    there     is    no 

greatest.      Let  that  number  a^  which  has  the  least 

index  of  those  greater  than  a^  be  a'p.      Similarly, 

let  a  p  be  that  number  of  the  series  {a'^}  which  has  the 

least  index  of  those  which  are  greater  than  a  ^ .      By 

proceeding  in  this  way,  we  get  ani  infinite  series  of 

increasing  numbers,  a  fundamental  series  in  fact, 

/ft  I 

a  1,  Op^,  Qp^,  .  .  .,  Op^,  .  .  . 

[226]   We  have 

I  <  /)2  <  Pa  <  •  •  '  <pv<  pv+i .  .  . , 
a\  <  a'p^  <  ap^  <  .  .  .  <  a^^  <  Up^^^  .  .  . , 

a\<ap^     always  if     /ui<p,; 

and  since  obviously  v  ^  p^,,  we  always  have 

a,  <  a'p^. 

Hence  we  see  that  every  number  a'^,  and  conse- 
quently every  number  a  <a,  is  exceeded  by  numbers 
a'p  for  sufficiently  great  values  of  v.  But  a  is  the 
number  which,  in  respect  of  magnitude,  immediately 
follows  all  the  numbers  a\  and  consequently  is  also 
the  next  greater  number  with  respect  to  all  ap  .  If, 
therefore,  we  put  a\  =  a-^,  Op^^^  =  «(,+!,  we  have 

a=  Lim  a^. 

V 

From    the    theorems  B,  C,  .  .  .,  K  it   is    evident 
that  the  numbers  of  the  second  number-class  result 


OF  TRANSFINITE  NUMBERS        169 

from  smaller  numbers  in  two  ways.  Some  numbers, 
which  we  call  "numbers  of  the  first  kind  {Art),''  are 
got  from  a  next  smaller  number  a_i  by  addition  of  1 
according  to  the  formula 

a  =  a-i+  I  ; 

The  other  numbers,  which  we  call  ' '  numbers  of  the 
second  kind,"  are  such  that  for  any  one  of  them 
there  is  not  a  next  smaller  number  a_i,  but  they 
arise  from  fundamental  series  [a^]  as  limiting 
numbers  according  to  the  formula 

a=  Lim  a^. 

V 

Here  a  is  the  number  which  follows  next  in  order 
of  magnitude  to  all  the  numbers  a^. 

We  call  these  two  ways  in  which  greater  numbers 
proceed  out  of  smaller  ones  ' '  the  first  and  the 
second  principle  of  generation  of  numbers  of  the 
second  number-class."* 


§  16 

The  Power  of  the  Second  Number- Class  is  equal 
to  the  Second  Greatest  Transfinite  Cardinal 
Number  Aleph-One 

Before  we  turn  to  the  more  detailed  considera- 
tions in  the  following  paragraphs  of  the  numbers  of 
the  second  number-class  and  of  the  laws  which 
rule  them,    we  will  answ^er  the  question  as  to  the 

*  [Cf.  Section  VII  of  the  Introduction,] 


I/O     THE  FOUNDING  OF  THE   THEORY 

cardinal  number  which  is  possessed  by  the  aggregate 
Z(h?o)={a}  of  all  these  numbers. 

[227]  A.  The  totality  {a}  of  all  numbers  a  of 
the  second  number-class  forms,  when  arranged  in 
order  of  magnitude,    a  well-ordered  aggregate. 

Proof. — If  we  denote  by  A„  the  totality  of 
numbers  of  the  second  number-class  which  are 
smaller  than  a  given  number  a,  arranged  in  order 
of  magnitude,  then  A„  is  a  well-ordered  aggregate 
of  type  a  — o).  This  results  from  theorem  H  of  §  14. 
The  aggregate  of  numbers  d  of  the  first  and  second 
number-class  which  was  there  denoted  by  {a'},  is 
compounded  out  of  {v]  and  A<^,  so  that 


Thus 
and  since 
we  have 


{a1  =  ({.},  A  J. 
{7}  =  M  +  A,; 

A„  =  a  —  w. 


Let  J  be  any  part  of  {«}  such  that  there  are 
numbers  in  {a}  which  are  greater  than  all  the 
numbers  of  J.  Let,  say,  a  be  one  of  these  numbers. 
Then  J  is  also  a  part  of  A^^+i,  and  indeed  such  a 
part  that  at  least  the  number  a  of  A^^+i  is  greater 
than  all  the  numbers  of  J.  Since  A^^+i  is  a  well- 
ordered  aggregate,  by  §  12  a  number  d  of  A^^+i, 
and  therefore  also  of  {a},  must  follow  next  to  all 
the  numbers  of  J.      Thus  the  condition  II  of  §  12  is 


OF  TRANSFINITE  NUMBERS         171 

fulfilled  in  the  case  of  {a}  ;  the  condition   1   of  §   12 
is  also  fulfilled  because  {a}  has  the  least  number  co. 

Now,  if  we  apply  to  the  well-ordered  aggregate 
{a}  the  theorems  A  and  C  of  §  12,  we  get  the 
following  theorems  : 

B.  Every  totality  of  different  numbers  of  the  first 
and  second  number-classes  has  a  least  number. 

C.  Every  totality  of  different  numbers  of  the  first 
and  second  number-classes  arranged  in  their  order  of 
magnitude  forms  a  well-ordered  aggregate. 

We  will  now  show  that  the  power  of  the  second 
number-class  is  different  from  that  of  the  first,  which 

is  No- 

D.  The  power  of  the  totality  {a}  of  all  numbers 
a  of  the  second  number-class  is  not  equal  to  ^^q. 

Proof. — If  {a}  were  equal  to  j^q,  we  could  bring 
the  totality  {a}  into  the  form  of  a  simply  infinite 
series 

Vp  y2>  •  •  •>  y.^.  •  •  • 

such  that  {y^}  would  represent  the  totality  of 
numbers  of  the  second  [228]  number-class  in  an 
order  which  is  different  from  the  order  of  magni- 
tude, and  {y^}  would  contain,  like  {a},  no  greatest 
number. 

Starting  from  y^,  let  y^  be  the  term  of  the  series 
which  has  the  least  index  of  those  greater  than  y^, 
yp  the  term  which  has  the  least  index  of  those 
greater  than  y^,  and  so  on.  We  get  an  infinite 
series  of  increasing  numbers, 

yi>  yp>  •  •  •'  ypv'  •  •  •> 


1/2     THE  FOUNDING  OF  THE   THEORY 
such  that 

yi<yp.<yp.r  •  •  <rp„<rp,+i<  •  •  •> 
r^  ^  yp,- 

By  theorem  C  of  §  15,  there  would  be  a  definite 
number  S  of  the  second  number-class,  namely, 

^  =  Limy,^, 

V 

which  is  greater  than  all  numbers  y^  .      Consequently 

we  would  have 

S>y. 

for  every  v.  But  {y^}  contains  all  numbers  of  the 
second  number-class,  and  consequently  also  the 
number  S  ;  thus  we  would  have,  for  a  definite  v^, 

^=yv 

which  equation  is  inconsistent  with  the  relation 
^  >  y^  .  The  supposition  {a}  =  t^o  consequently  leads 
to  a  contradiction. 

E.  Any  totality  {/3}  of  different  numbers  ^  of 
the  second  number-class  has,  if  it  is  infinite,  either 
the  cardinal  number  «(,  or  the  cardinal  number  {a} 
of  the  second  number-class. 

Proof. — The  aggregate  {^},  when  arranged  in  its 
order  of  magnitude,  is,  since  it  is  a  part  of  the  well- 
ordered  aggregate  {a},  by  theorem  O  of  §  13, 
similar  either  to  a  segment  Aa  ,  which  is  the  totality 


OF  TRANSFINITE  NUMBERS        173 

of  all  numbers  of  the  same  number-class  which  are 
less  than  a©,  arranged  in  their  order  of  magnitude, 
or  to  the  totality  {a}  itself.  As  was  shown  in  the 
proof  of  theorem  A,  we  have 

Thus  we  have  either  {^)—a^  —  w  or   {/3}  =  {a},  and 


consequently  {^}  is  either  equal  to  Qq  — w  or  {a). 
But  oq  — ft)  is  either  a  finite  cardinal  number  or  is 
equal  to  «o  (theorem  I  of  §  15).  The  first  case  is 
here  excluded  because  {/3}  is  supposed  to  be  an 
infinite  aggregate.  Thus  the  cardinal  number  {^} 
is  either  equal  to  j^q  or  {a}. 

F.  The  power  of  the  second  number-class  {a}  is 
the  second  greatest  transfinite  cardinal  number 
Aleph-one. 

[229]   Proof. — There    is    no    cardinal    number    a 

which  is  greater  than  n^  and  less  than  {a}.  For  if 
not,  there  would  have  to  be,  by  §  2,  an  infinite  part 

{/3}  of  {a}  such  that  {/3}=a.  But  by  the  theorem 
E  just  proved,  the  part  {/3}  has  either  the  cardinal 

number  js^q  or  the  cardinal  number  {a}.  Thus  the 
cardinal  number  {a\  is  necessarily  the  cardinal 
number  which  immediately  follows  «o  in  magnitude  ; 
we  call  this  new  cardinal  number  «j. 

In  the  second  number-class  Z(«o)  we  possess, 
consequently,  the  natural  representative  for  the 
second  greatest  transfinite  cardinal  number  Aleph- 
one. 


174     THE  FOUNDING  OF  THE   THEORY 

%  17 

The  Numbers  of  the  Form  a)%  +  a)'""\+  .  .  .  +j/^. 

It  is  convenient  to  make  ourselves  familiar,  in  the 
first  place,  with  those  numbers  of  Z({^o)  which  are 
whole  algebraic  functions  of  finite  degree  of  w. 
Every  such  number  can  be  brought — and  brought 
in  only  one  way — into  the  form 

(I)  0  =  a,%  +  a)'^-V+  .  .  .    +.^, 

where  ;x,  v^  are  finite  and  different  from  zero,    but 
j^i,  1^2'  •  •  •>  '^i'*  "^^y  ^^  zero.      This  rests  on  the  fact 


4,  we 


and,  by  theorem  E  of  §  15, 

Thus,  in  an  aggregate  of  the  form 

.  .  .  +w'^V  +  w^i/+  .  .  ., 

all  those  terms  which  are  followed  towards  the  right 
by  terms  of  higher  degree  in  «  may  be  omitted. 
This  method  may  be  continued  until  the  form  given 
in  (i)  is  reached.      We  will  also  emphasize  that 


that 

(2) 

of'v  -\r 

Iffv  =  61)' 

V, 

if  m'</x 

and 

i/>0,  />o. 

For, 

by  (8) 

of 

have 

iJd^  V 

+  ^^^1/  = 

i/{v^. 

^^-^v\ 

(3)  w'^J/  +  ft)V  =  a)'^(t/  +  0- 


OF  TRANSFINITE  NUMBERS        175 

Compare,  now,  the  number  0  with  a  number  \p-  of 
the  same  kind: 

(4)  V^  =  coVo  +  ^')^"Vi+  •  •  •  +P^' 

If  yu  and  X  are  different  and,  say,  iu<Xy  we  have  by 
(2)  0  +  \^  =  -v/r,  and  therefore  (p<\fr. 

[230]  If  jUL  =  \,  vq,  and  Pq  are  different,  and,  say, 
i'o<Poy  ^^^  h^^'^  t)y  (2) 

0  +  (^Xpo-^o)  +  ^^"Vi+  •  •  •  +M  =  \^i 
and  therefore 

If,  finally, 

/x  =  X,      V(i  =  Po,  I'l  —  Pv  •  '  •  ^<T-l  =  PcT-l^      (rKfXy 

but   i;^   is   different   from   p^  and,    say,    v^<p„,    we 
have  by  (2) 

and  therefore  again 

Thus,  we  see  that  only  in  the  case  of  complete 
identity  of  the  expressions  ^  and  yp-  can  the  numbers 
represented  by  them  be  equal. 

The  addition  of  0  and  \/r  leads  to  the  following 
result  : 

{a)  If  ^<A,  then,  as  we  have  remarked  above, 

{b)  If  ^  =  X,  then  we  have 


176     THE  FOUNDING  OF  THE   THEORY 
{c)  \i  fx>\  we  have 

In  order  to  carry  out  the  multiplication  of  0  and  i/r, 
we  remark  that,  if  p  is  a  finite  number  which  is 
different  from  zero,  we  have  the  formula  : 

(5)  ^yo  =  a)%/D  +  w'^-Vi+  ...  +1/^. 

It  easily  results  from  the  carrying  out  of  the  sum 
consisting  of  p  terms  0  +  ^+  .  .  .  +0.  By  means 
of  the  repeated  application  of  the  theorem  G  of 
§  15  we  get,  further,  remembering  the  theorem  F 
of  §15, 

(6)  0CO  =  ft)'^+S 

and  consequently  also    . 

(7)  9^0)^  =  60^  +  ^. 

By  the  distributive  law,  numbered  (8)  of  §  14, 
we  have 

0\/^  =  0a)^iOo  +  0w^'Vi+  •  •  •  +V^^Px-i  +  0/3a. 

Thus  the  formulae  (4),  (5),  and  (7)  give  the  following 
result  : 

{a)  if  p;^  =  o,  we  have 

{b)  If  Pa  is  not  equal  to  zero,  we  have 


OF  TRANSFINITE  NUMBERS        177 

[231]  We  arrive  at  a  remarkable  resolution  of 
the  numbers  0  in  the  following  manner.      Let 

(8.)  0  =  a)%  +  a)'^'/ci+  .  .  .   -^(*r--K,, 

where 

and  /Cq,  /ci,  .  .  .,  act  3-re  finite  numbers  which  are 
different  from  zero.      Then  we  have 

^  =  (co'^'/ci  +  w^^/ca  +  .  .  .  +  ur-^K,){s^^  -  '^•/f 0  +  I ). 

By  the  repeated  application  of  this  formula  we  get 

0  =  fo'^r/c^(w'"T-l-MT/f^_^+  l)(ft)'^T-2-'^T_l^^_2+  l).    .    . 

(co'^-'^'/Co+l). 
But,  now, 

a)\+  I  =(co^+  l)/c, 

if  /c  is  a  finite  number  which  is  different  from  zero ; 
so  that : 

.   .   .   (ft)'^-'^'+I>o. 

The  factors  0)^+  i  which  occur  here  are  all  irre- 
soluble,  and  a  number  0  can  be  represented  in  this 
product-form  in  only  one  way.  If  fXj  —  o^  then  0 
is  of  the  first  kind,  in  all  other  cases  it  is  of  the 
second  kind. 

The  apparent  deviation  of  the  formulae  of  this 
paragraph  from  those  which  were  given  in  Math. 
Ann.,  vol.  xxi,  p.  585  (or  Grundlagen,  p.  41),  is 
merely  a  consequence  of  the  different  writing  of  the 
product  of   two    numbers  :    we  now  put  the  multi- 

12 


178     THE  FOUNDING  OF  THE   THEORY 

plicand    on    the    left    and    the   multipHcator  on  the 
right,  but  then  we  put  them  in  the  contrary  way. 


§i8 

The  Power  *  y*  in  the  Domain  of  the  Second 
Number- Class 

Let  ^  be  a  variable  whose  domain  consists  of  the 
numbers  of  the  first  and  second  number-classes  in- 
cluding zero.  Let  y  and  (5  be  two  constants  belong- 
ing to  the  same  domain,  and  let 

^>0,      y>l. 
We  can  then  assert  the  following  theorem  : 

A.  There  is  one  wholly  determined  one-valued 
function /(^)  of  the  variable  ^  such  that  : 

{a)  /(o)  =  ^. 

{b)  \i  ^'  and  ^'  are  any  two  values  of  ^,  and  if 

then 

/(f)</(f)- 

[232]   ic)  For  every  value  of  ^  we  have 

/(#+i)=/(ar- 

{d)  If  {^^}  is  any  fundamental  series,  then  {/(f^)} 
is  one  also,  and  if  we  have 

^=Lim  ^„ 

V 

then 

/(f)  =  Lini/(,^„). 

V 

*  [Here  obviously  it  is  Potenz  and  not  MdchHgkeit.'\ 


OF  TRANSFINITE  NUMBERS        179 

Proof. — By  {a)  and  {c),  we  have 

/(l)  =  <5y,     f{2)  =  Syy,     f{z)  =  Syyy,     .  .  ., 
and,  because  5>0  and  y>  i,  we  have 

/(l)</(2)</(3)<  .  .  .  </(„)</(.+  I)<  .  .  . 

Thus  the  function  f{^)  is  wholly  determined  for  the 
domain  $<w.  Let  us  now  suppose  that  the  theorem 
is  valid  for  all  values  of  f  which  are  less  than  a, 
where  a  is  any  number  of  the  second  number-class, 
then  it  is  also  vaUd  for  f  <a.  For  if  a  is  of  the 
first  kind,  we  have  from  {c)  : 

/(a)=/(a-Oy>/(a.i); 

SO  that  the  conditions  {b),  (c),  and  (d)  are  satisfied 
for  f^a.  But  if  a  is  of  the  second  kind  and  {a^}  is 
a  fundamental  series  such  that   Lim  a^  =  a,  then  it 

follows  from  (d)  that  also  {/(ay)]  is  a  fundamental 
series,   and  from  (d)  that /(a)  =  Lim /(a^).      If  we 

take  another  fundamental  series  {a^}  such  that 
Lim  a\,  =  a,    then,    because   of  (d),    the   two   funda- 

mental  series  {/(a,,)}  and  {/(a\)}  are  coherent,  and 
thus  also /(a)  =  Lim /(a'„).      The   value  of /(«)  is, 

consequently,  uniquely  determined  in  this  case  also. 
If  a'  is  any  number  less  than  a,  we  easily  convince 
ourselves  that  /(a) < /(a).  The  conditions  (d),  (c)y 
and  (d)  are  also  satisfied  for  ^<'a.  Hence  follows 
the  validity  of  the  theorem /<?r  all  values  of  ^.  For 
if  there  were  exceptional  values  of  f  for  which  it 
did  not  hold,  then,   by  theorem   B  of  §  16,  one  of 


i8o     THE  FOUNDING  OF  THE   THEORY 

them,  which  we  will  call  a,  would  have  to  be  the 
least.  Then  the  theorem  would  be  valid  for  f<a, 
but  not  for  ^^a,  and  this  would  be  in  contradiction 
with  what  we  have  proved.  Thus  there  is  for  the 
whole  domain  of  f  one  and  only  one  function  /(f) 
which  satisfies  the  conditions  {a)  to  {d). 

[233]  If  we  attribute  to  the  constant  ^  the  value  i 
and  then  denote  the  function  f{£)  by 

we  can  formulate  the  following  theorem  : 

B.  If  y  is  any  constant  greater  than  i  Which 
belongs  to  the  first  or  second  number-class,  there 
is  a  wholly  definite  function  y^  of  f  such  that  : 

{a)  y«=i; 

W  If  f  <rthen  y^'<y^"; 
{c)  For  every  value  of  f  we  have  y^+i  =  y^y  ; 
id)  If   {f^}    is   a   fundamental   series,    then    {y^"} 
is   such   a   series,   and  we   have,    if  f=Lim  f^,    the 

V 

equation 

y^  =  Lim  y^". 

V 

We  can  also  assert  the  theorem  : 

C.  If /(f)  is  the  function  of  f  which  is  characterized 
in  theorem  A,  we  have 

/(f) =V- 

Proof. — If  we  pay  attention  to  (24)  of  §  14, 
we  easily  convince  ourselves  that  the  function  ^y^ 
satisfies,  not  only  the  conditions  {a),  {b),  and  {c) 
of  theorem  A,    but   also   the   condition   {d)  of  this 


OF  TRANSFINITE  NUMBERS        i8i 

theorem.       On    account   of   the   uniqueness   of  the 
function /(f),  it  must  therefore  be  identical  with  ^y^. 
D.    If  a  and  ^  are  any  two  numbers  of  the  first 
or  second  number-class,  including  zero,  we  have 

y»+^  =  y*y^. 

Proof.— W&    consider    the    function   0(f)  =  y«+^. 
Paying  attention  to  the  fact  that,  by  formula  (23) 

of  §  14, 

Lim  (a  +  f,)  =  a  +  Lim  f„ 

V  V 

we  recognize  that  ^(f)  satisfies  the  following  four 
conditions  : 

{a)  0(0)  =  y*; 

{b)  Iff  <r,  then  0(f)  <0(r); 

{c)  For  every  value  of  f  we  have  V'(f  +  0  =  <^(f  )y  5 

(d)  If    {f }    is    a    fundamental    series    such    that 
Lim  f  =  f ,  we  have 

0(f)  =  Lim  0(f). 

V 

By  theorem  C  we  have,  when  we  put  (5  =  y^ 

0(a  =  yy- 

If  we  put  f  =  ^  in  this,  we  have 

E.    If  a  and  /3  are  any  two  numbers  of  the  first  or 
second  number-class,  including  zero,  we  have 

[234]   Proof. — Let     us     consider    the     function 
■^(f)  =  y«f    and   remark   that,    by  (24)   of  §    14,    we 


1 82     THE  FOUNDING  OF  THE   THEORY 
always   have    Lim  af^  =  a  Lim  ^^,   then  we   can,    by 

V  V 

theorem  D,  assert  the  following  : 

{a)  iA(o)=i  ; 

ib)  Iff<r,  thenV^(0<V^(r); 

ic)  For  every  value  of  f  we  have  ^^(f  +  i )  =  V^(f)y*  ; 

{d)  If  {^^}  is  a  fundamental  series,  then  {V^(^v)}  is 
also  such  a  series,  and  we  have,   if  ^=Lim^j,,  the 

V 

equation  '\/r(^)  =  Lim  ■^{^^. 

V 

Thus,  by  theorem  C,  if  we  substitute  in  it  i  for  (5 
and  y*  for  y, 

On  the  magnitude  of  y^  in  comparison  with  ^  we 
can  assert  the  following  theorem  : 

F.    If  y  >  I,  we  have,  for  every  value  of  f, 

Proof. — In  the  cases  ^  =  o  and  ^=  I  the  theorem 
is  immediately  evident.  We  now  show  that,  if  it 
holds  for  all  values  of  ^  which  are  smaller  than  a 
given  number  a>  I,  it  also  holds  for  ^=a. 

If  a  is  of  the  first  kind,  we  have,  by  supposition, 

a_i<y«-S 
and  consequently 

«-iy  <  y''"'y  =  r''- 

Hence 

y'^>a_i  +  a_i(y-l). 

Since  both  a_^  and  y—  i  are  at  least  equal  to  i,  and 
a_i+  I  =a,  we  have 

y"  >  a. 


OF  TRANSFINITE  NUMBERS        183 

If,  on  the  other  hand,  a  is  of  the  second  kind  and 
a  =  Lim  a„, 


then,  because  a^,<a,  we  have  by  supposition 

Consequently 

Lim  a„  S.  Lim  y"", 

V  V 

that  is  to  say, 

a  <  y*. 

If,  now,  there  were  values  of  ^  for  which 

one  of  them,  by  theorem  B  of  §  16,  would  have  to 
be  the  least.  If  this  number  is  denoted  by  a,  we 
would  have,   for  ^<a, 

[235]  i^y'; 

but 

a>y% 

which  contradicts  what  we  have  proved  above. 
Thus  we  have  for  all  values  of  ^ 

§  19 

The  Normal  Form  of  the  Numbers  of  the 

Second  Number- Class 

Let  a  be  any  number  of  the  second  number-class. 
The    power  00^  will  be,  for  sufficiently  great  values 


1 84     THE  FOUNDING  OF  THE   THEORY 

of  ^,  greater  than  a.  By  theorem  F  of  §  i8,  this  is 
always  the  case  for  f  >a  ;  but  in  general  it  will  also 
happen  for  smaller  values  of  f. 

By  theorem  B  of  §  i6,  there  must  be,  among  the 
values  of  ^  for  which 

d)^>  a, 

one  which  is  the  least.  We  will  denote  it  by  /3,  and 
we  easily  convince  ourselves  that  it  cannot  be  a 
number  of  the  second  kind.      If,  indeed,  we  had 

/5  =  Lim  /5j,, 

V 

we  would  have,  since  ^^  <  /3, 

CO  "  <  a, 

and  consequently 

Lim  0)^'  ^  a. 

V 

Thus  we  would  have 

o)^  <  a, 
whereas  we  have 

Therefore  /3  is  of  the  first  kind.  We  denote  /3_i 
by  Oq,  so  that  ^  =  ao+i,  and  consequently  can 
assert  that  there  is  a  wholly  determined  number  a^ 
of  the  first  or  second  number-class  which  satisfies 
the  two  conditions  : 

(l)  (jo^<^a,      o)'^o)>a. 

From  the  second  condition  we  conclude  that 


OF  TRANSFINITE  NUMBERS         185 

does  not  hold  for  all  finite  values  of  v,  for  if  it  did 
we  would  have 

Lim  a)%  =  co"«a)  <  a. 

V 

The  least  finite  number  v  for  which 
{£)  ^v>  a 

will  be  denoted  hy  kq-\-i.  Because  of  (i),  we  have 
/Co  >  o. 

[236]   There  is,   therefore,   a  wholly  determined 
number  k^  of  the  first  number-class  such  that 

(2)  a)''oA:o< a,      a)%(/Co  +  l)  >  a. 
If  we  put  a  — a)%/co  =  a',  we  have 

(3)  a  =  a)%/Co  +  a' 
and 

(4)  0<a'<a)%,      0</fo<a). 

But  a  can  be  represented  in  the  form  (3)  under  the 
conditions  (4)  in  only  a  single  way.  For  from  (3) 
and  (4)  follow  inversely  the  conditions  (2)  and  thence 
the  conditions  (i).  But  only  the  number  ao=|8_i 
satisfies  the  conditions  (i),  and  by  the  conditions 
(2)  the  finite  number  /Cq  is  uniquely  determined. 
From  (i)  and  (4)  follows,  by  paying  attention  to 
theorem  F  of  §  18,  that 

a  <a,      Oo^a. 

Thus  we  can  assert  the  following  theorem  : 

A.    Every  number  a  of  the  second  number-class 


1 86     THE  FOUNDING  OF  THE   THEORY 

can  be  brought,  and  brought  in  only  one  way,  into 
the  form 

a  =  w^o/Cq  +  a  , 
where 

O  <.a'  <  co%,      o  <  /Co  <  Wj 

and  a    is  always  smaller  than  a,  but  a^  is   smaller 
than  or  equal  to  a. 

If  a  is  a  number  of  the  second  number-class,  we 
can  apply  theorem  A  to  it,  and  we  have 

(5)  a' =  coVi  +  a'', 

where 

0<^a   <i>f-h      0<Ki<(j», 
and 

ai  <  Oq,      a^  <  a. 

In  general  we  get  a  further  sequence  of  analogous 
equations  : 

(6)  a'  =  of-'^K2  +  a^'j 

(7)  a'''  =  a)'^3/c3  +  a^\ 


But  this  sequence  cannot  be  infinite,  but  must 
necessarily  break  off.  For  the  numbers  a,  a,  a\  .  .  . 
decrease  in  magnitude  : 


a>  a>  a'>  a"  .  .  . 


If  a  series  of  decreasing  transfinite  numbers  were 
infinite,  then  no  term  would  be  the  least  ;  and  this 
is  impossible  by  theorem  B  of  §  i6.  Consequently 
we  must  have,  for  a  certain  finite  numerical  value  r. 


«^^+^;=o- 


OF  TRANSFINITR  NUMBERS         187 

[237]  If  we  now  connect  the  equations  (3),  (5), 
(6),  and  (7)  with  one  another,  we  get  the  theorem  : 

B.  Every  number  a  of  the  second  number-class 
can  be  represented,  and  represented  in  only  one 
way,  in  the  form 

where  Oq,  ai  ,  .  .  .  a^  are  numbers  of  the  first  or 
second  number-class,  such  that  : 

a^  >  Oj  >  ag  >  .  .  .  >  ctj  ^  O, 

while  /Co,  /ci,  .  .  .  /c^,  r+i  are  numbers  of  the  first 
number-class  which  are  different  from  zero. 

The  form  of  numbers  of  the  second  number-class 
which  is  here  shown  will  be  called  their  ' '  normal 
form "  ;  a^  is  called  the  ' '  degree "  and  a^  the 
''exponent"  of  a.  For  t  =  o,  degree  and  exponent 
are  equal  to  one  another. 

According  as  the  exponent  a^  is  equal  to  or  greater 
than  zero,  a  is  a  number  of  the  first  or  second  kind. 

Let  us  take  another  number  ^  in  the  normal 
form  : 

(8)  ^  =  a)^«Xo  +  a)^'Xi+  .  •  .   +co^'^Xa. 
The  formulae  : 

(9)  w'^'/c'  -I-  w'^'/c  =  w'^X'c'  +  /c), 

(10)  ■     a)*"'/ +  co*V' =  w'^V,      a  <d\ 

where  /c,  k\  k'  here  denote  finite  numbers,  serve 
both   for   the  comparison  of  a  with  ^  and  for  the 


1 88     THE  FOUNDING  OF  THE   THEORY 

carrying  out  of  their  sum  and  difference.      Tiiese  are 
generalizations  of  the  formulae  (2)  and  (3)  of  §  17. 

For  the  formation  of  the  product  a^,  the  following 
formulae  come  into  consideration  : 


(11)     aX  =  (o'^^X  +  w-^'/ci  +  .  .  .   +w*X,      0<X< 
(12) 


Od  , 


=  /.>«o+l 


aa)=  o) 


(13)  aw^' =  a)'^o+^;      /3'>0. 

The  exponentiation  a^  can  be  easily  carried  out 
on  the  basis  of  the  following  formulae  : 

(14)  a^  =  a)*o%+  .  .  .,      0<X<a). 

The  terms  not  written  on  the  right  have  a  lower 
degree  than  the  first.  Hence  follows  readily  that 
the  fundamental  series  {a^}  and  {co'"^^}  are  coherent, 
so  that 

(15)  a"  =  0)^0-,      ao>0. 

Thus,  in  consequence  of  theorem  E  of  §  18,  we 
have  : 

(16)  a"^'  =  w"''"^      ao>0,      ^'>0. 

By  the  help  of  these  formulae  we  can  prove  the 
following  theorems  : 

[238]  C.  If  the  first  terms  w^o/cq,  w^oXq  of  the 
normal  forms  of  the  two  numbers  a  and  /3  are  not 
equal,  then  a  is  less  or  greater  than  /3  according  as 
a)«o/c-Q  is  less  or  greater  than  co^^Xq.      But  if  we  have 

and  if  w'^'^+Vp+i  is  less  or  greater  than  w^p-^\+i,  then 
a  is  correspondingly  less  or  greater  than  ^. 


OF  TRANSFINITE  NUMBERS         189 

D.  If  the  degree  a^  of  a  is  less  than  the  degree 
i8o  of  ^,  we  have 

a  +  /3  =  /3. 
If  ao  =  /5o,  then 

a  +  /3  =  a)^o(;,^  +  Xo)  +  a)^*Xi+  .  .  .   +oAA,. 
But  if 

then 

a  +  ^  =  w'^/CoH-  .  .  .  +a)"''/Cp  +  a)^«Xo  +  ^'^i+  •  •  •  +^"X^- 

E.  If  ^  is  of  the  second  kind  (/3^>o),  then 
a/3  =  (o'^o+^oXQ  +  a)«o+^^Xi+  .  .  .   +  w'^o+^-X^  =  w*o/3  ; 

But  if /3  is  of  the  first  kind  (/3,  =  o),  then 

a^  =  (o«o+^oX(,  +  a)«^^+^^Xi+  .  .  .+co'^'>+^'--iX^_i  +  oj'^*'/CoX^ 

4-  w'^'/ci  +  .  .  .  +  o/^/c^. 

F.  If  /3  is  of  the  second  kind  (/3<^>o),  then 

But  if  (3  is   of  the    first  kind  (^^  =  0),   and   indeed 
8  =  ^'  +  Xo-,  where  (3'  is  of  the  second  kind,  we  have: 

G.  Every  number  a  of  the  second  number-class 
can  be  represented,  in  only  one  way,  in  the  product- 
form  : 

a  =  a)V(w^'+  l)Kr-lW'+  l)/Cr-2  •    •    •    (o)^"  +  iK, 

and  we  have 

70  =  «T)  yi  =  ar-i  — aT5  y2  =  Wr-2  — «T-1,  .  .  •>yT  =  «o'~«i' 


190     THE  FOUNDING  OF  THE   THEORY 

whilst  /cq,  /cj,  .  .  .  K^  have  the  same  denotation  as 
in  the  normal  form.  The  factors  (0"^+!  are  all 
irresoluble. 

H.  Every  number  a  of  the  second  kind  which 
belongs  to  the  second  number-class  can  be  repre- 
sented, and  represented  in  only  one  way,  in  the 
form 

a  —  L'd'^a  i 

where  yo>o  and  a  is  a  number  of  the  first  kind 
which  belongs  to  the  first  or  second  number-class. 

[239]  I.  In  order  that  two  numbers  a  and  ^  of 
the  second  number-class  should  satisfy  the  relation 

it  is  necessary  and  sufficient  that  they  should  have 
the  form 

a  =  yyw,     /3  =  yv^ 

where  /x  and  v  are  numbers  of  the  first  number-class. 
K.    In    order    that    two  numbers  a  and  ^  of  the 
second    number-class,   which    are  both  of   the    first 
kind,  should  satisfy  the  relation 

it  is  necessary  and  sufficient  that  they  should  have 
the  form 

where  ^  and  v  are  numbers  of  the  first  number-class. 

In  order  to  exemplify  the  extent  of  the  normal 

form  dealt  with  and  the  product-form  immediately 

connected    with    it,    of  the  numbers    of  the  second 


OF  TRANSFINITE  NUMBERS        191 

number-class,  the  proofs,  which  are  founded  on 
them,  of  the  two  last  theorems,  I  and  K,  may  here 
follow. 

From  the  supposition 

a  +  /3  =  /3  +  a 

we  first  conclude  that  the  degree  oq  of  a  must  be 
equal  to  the  degree  ^^  of  ^.  For  if,  say,  Qq  </3o,  we 
would  have,  by  theorem  D, 

a  +  ^  =  A 
and  consequently 

which  is  not  possible,  since,  by  (2)  of  §  14, 

Thus  we  may  put 

a  =  (jo^fA  +  a',      |8  =  M^'ov  + 13', 

where  the  degrees  of  the  numbers  a  and  /3'  are  less 
than  ao,  and  ju  and  v  are  infinite  numbers  which  are 
different  from  zero.      Now,  by  theorem  D  we  have 

a  +  P  =  w''ifjL  +  v)  +  ^\     P  +  a-=co'^(^  +  p)  +  a, 

and  consequently 

(^''Klii  + 1/)  +  /3'  =  co'^ifjL  + 1/)  +  a. 

By  theorem  D  of  §  14  we  have  consequently 

13' =  a. 
Thus  we  have 

a  =  (jo"-"/ul  -\-a,      13  =  iff--v  +  a\ 


192     THE  FOUNDING  OF  THE   THEORY 

[240]  and  if  we  put 

uf-''  +  a'  =  7 
we  have,  by  (11): 

a  =  y/x,      /5  =  yv. 

Let  us  suppose,  on  the  other  hand,  that  a  and  ^  are 
two  numbers  which  belong  to  the  second  number- 
class,  are  of  the  first  kind,  and  satisfy  the  condition 

and  we  suppose  that 

We  will  imagine  both  numbers,  by  theorem  G,  in 
their  product-form,  and  let 

where  a  and  /3'  are  without  a  common  factor  (besides 
i)  at  the  left  end.      We  have  then 

a>/3\ 
and 

All  the  numbers  which  occur  here  and  farther  on 
are  of  the  first  kind,  because  this  was  supposed  of 
a  and  /3. 

The  last  equation,  when  we  refer  to  theorem  G, 
shows  that  a  and  /3'  cannot  be  both  transfinite, 
because,  in  this  case,  there  would  be  a  common 
factor  at  the  left  end.  Neither  can  they  be  both 
finite  ;  for  then  S  would  be  transfinite,  and,  if  k  is 
the  finite  factor  at  the  left  end  of  S,  we  would  have 


OP  TRANSPINITE  NUMBERS        193 
and  thus 

Thus  there  remains  only  the  possibility  that 

a  >o),     P'  <  CO. 
But  the  finite  number  /3'  must  be  t  : 

because  otherwise  it  would  be  contained  as  part  in 
the  finite  factor  at  the  left  end  of  «'. 

We  arrive  at  the  result  that  /3  =  (5,  consequently 

a  =  (3a  , 

where  a  is  a  number  belonging  to  the  second 
number-class,  which  is  of  the  first  kind,  and  must 
be  less  than  a  : 

a  <a. 

Between  a  and  ^  the  relation 

afi  =  l3a 
subsists. 

[241]  Consequently  if  also  a>l3,  we  conclude  in 
the  same  way  the  existence  of  a  transfinite  number 
of  the  first  kind  a'  which  is  less  than  a  and  such  that 

a=Pa\     a '13  = /3a'. 

If  also  a'  is  greater  than  (3,  there  is  such  a  number 
a"'  less  than  a\  such  that 

a   =pa    ,     a   p  =  /5a    , 

and  so  on.  The  series  of  decreasing  numbers,  a,  a', 
a\   a"\  .  .  .,    must,    by  theorem    B    of  §    16,   break 


\ 


i3 


194     THE  FOUNDING  OF  THE   THEORY 

off.      Thus,   for  a  definite  finite  index  ^q,  we  must 
have 


(p.  < 


a^'^o 


/3. 


If 

we  have 

the  theorem  K  would  then  be  proved,  and  we  would 
have 

But  if 

then  we  put 

and  have 

Thus  there  is  also  a  finite  number  p^  such  that 

In  general,  we  have  analogously  : 

and  so  on.  The  series  of  decreasing  numbers  /3i, 
/^a,  I3q,  .  .  .  also  must,  by  theorem  B  of  §  i6,  break 
off.      Thus  there  exists  a  finite  number  k  such  that 


If  we  put 
then 


OF  TRANSFTNITR  NUMBERS        195 

where  /x  and  v  are  numerator  and  denominator  of 
the  continued  fraction: 


+- 
p. 


[242]  §   20 

The  e-Numbers  of  the  Second  Number- Class 

The  degree  a©  of  a  number  a  is,  as  is  immediately 
evident  from  the  normal  form  : 

(1)  a  =  w'^o/Cq  +  (o*i/Ci  +  .  .  . ,    «(,  >  ai  >  .  .  . ,    0<k„<w, 

when  we  pay  attention  to  theorem  F  of  §  18,  never 
greater  than  a  ;  but  it  is  a  question  whether  there 
are  not  numbers  for  which  a^  =  a.  In  such  a  case 
the  normal  form  of  a  would  reduce  to  the  first  term, 
and  this  term  would  be  equal  to  w*,  that  is  to  say, 
a  would  be  a  root  of  the  equation 

(2)  a,^  =  f. 

On  the  other  hand,  every  root  a  of  this  equation 
would  have  the  normal  form  w" ;  its  degree  would 
be  equal  to  itself. 

The  numbers  of  the  second  number-class  which 
are  equal  to  their  degree  coincide,  therefore,  with 
the  roots  of  the  equation  (2).  It  is  our  problem  to 
determine  these  roots  in  their  totality.  To  dis- 
tinguish them  from  all  other  numbers  we  will  call 
them  the  "  f-numbers  of  the  second  number-class." 


196     THE  FOUNDING  OF  THE  THEORY 

That    there    are    such    e-numbers    results   from    the 
following  theorem  : 

A.  If  y  is  any  number  of  the  first  or  second 
number-class  which  does  not  satisfy  the  equation 
(2),  it  determines  a  fundamental  series  {y}  by  means 
of  the  equations 

71  =  ^^      y2  =  w^S      •  •  'J      y^  =  w^''"S      .  .  • 
The  limit  Lim  y^  =  E(y)  of  this  fundamental  series 

V 

is  always  an  e-number. 

Proof. — Since  y  is  not  an  e-number,  we  have 
a)>'>y,  that  is  to  say,  7i>y.  Thus,  by  theorem  B 
of  §  1 8,  we  have  also  w'^'i  >  w'>',  that  is  to  say,  y2>yi  ; 
and  in  the  same  way  follows  that  y3>y2,  and  so 
on.  The  series  {y^}  is  thus  a  fundamental  series. 
We  denote  its  limit,  which  is  a  function  of  y,  by 
E(y)  and  have  : 

^E(v)  _  Lin^  ^v^^  Lin^  y^^^  —  E(y). 

V  V 

Consequently  E(y)  is  an  e-number. 

B.  The  number  eo  =  E(i)  =  Lim  w^,,  where 

V 

is  the  least  of  all  the  e-numbers. 

[243]   Proof. — Let  e  be  any  e-number,  so  that 


Since  e'>w,  we  have  w''>w'",  that  is  to  say,  e  >  w^. 
Similarly  t/xo'*'',  that  is  to  say,  e'xo^,  and  so  on. 
We  have  in  general 


OF  TRANSFINITE  NUMBERS        197 

and  consequently 

e'>.Lim  a)„, 

that  is  to  say, 

Thus  eo  =  E(i)  is  the  least  of  all  e-numbers. 

C.    If  e'  is  any  e-number,   e"  is  the  next  greater 
e-number,  and  y  is  any  number  which  lies  between 

them  : 

e' <  y  <  e', 
then  E(y)  =  e". 
Proof. — From 

e'  <  y  <:  e' 
follows 

that  is  to  say, 

e'  <  yi  <  e' . 

Similarly  we  conclude 

e'  <  ya  <  e", 
and  so  on.      We  have,  in  general, 

e  <  y,,  <  e'', 
and  thus 

e'<E(y)<e". 

By  theorem  A,  E(y)  is  an  e-number.  Since  e'  is 
the  e-number  which  follows  e'  next  in  order  of  mag- 
nitude, E(y)  cannot  be  less  than  e",  and  thus  we 
must  have 

E(y)  =  e". 

Since  e'+i  is  not  an  e-number,  simpl}-  because  all 
6-numbers,  as  follows  from  the  equation  of  definition 


198     THE  FOUNDING  OF  THE   THEORY 

i  =  w^,  are  of  the  second  kind,  e -{-  i  is  certainly  less 
than  e\  and  thus  we  have  the  following  theorem  : 

D.  If  e'  is  any  e-number,  then  E(e'+  i)  is  the  next 
greater  e-number. 

To  the  least  e-number,  €q,  follows,  then,  the  next 
greater  one: 

ei  =  E(eo+i), 

[244]   to  this  the  next  greater  number  : 

^2  =  E(6i+l), 

and  so  on.  Quite  generally,  we  have  for  the 
(1/+  i)th  e-number  in  order  of  magnitude  the  formula 
of  recursion 

(3)  e,  =  E(e,_i-|-l). 

But  that  the  infinite  series 

^0'    ^1)    •    •    •    6,.,    .   .    . 

by  no  means  embraces  the  totality  of  e-numbers 
results  from  the  following  theorem  : 

E.  If   e,    e',    e',    ...    is    any     infinite     series     of 
e-numbers  such  that 

e<e'<e".  .  .  e^^^  <  e^-^+i^  <  .  .  ., 

then     Lim    e^''^    is    an  e-number,   and,    in    fact,    the 

I' 
e-number  which  follows  next  in  order  of  magnitude 
to  all  the  numbers  eH 
Proof.— 

Lim  eW  . ,) 

(0  "        =  Lim  w*    =  Lim  e^''^. 


OF  TRANSFINITE  NUMBERS         199 
That   Lim  e^''^  is  the  e-number  which  follows  next 

V 

in  order  of  magnitude  to  all  the  numbers  e^"^  results 
from  the  fact  that  Lim  e^"^   is   the   number   of  the 

second  number-class  which  follows  next  in  order  of 
magnitude  to  all  the  numbers  e^^^ 

F.  The  totality  of  e-numbers  of  the  second 
number-class  forms,  when  arranged  in  order  of 
magnitude,  a  well-ordered  aggregate  of  the  type  Vt 
of  the  second  number-class  in  its  order  of  magnitude, 
and  has  thus  the  power  Aleph-one. 

Proof. — The  totaHty  of  e-numbers  of  the  second 
number-class,  when  arranged  in  their  order  of  magni- 
tude, forms,  by  theorem  C  of  §  16,  a  well-ordered 
aggregate  : 

(4)         €q,  e^,  .  .  .,  e^,,  .  .  .  €0,+!,  .  .  .  ea'  .  .  ., 

whose  law  of  formation  is  expressed  in  the  theorems 
D  and  E.  Now,  if  the  index  a  did  not  successively 
take  all  the  numerical  values  of  the  second  number- 
class,  there  would  be  a  least  number  a  which  it  did 
not  reach.  But  this  would  contradict  the  theorem 
D,  if  a  were  of  the  first  kind,  and  theorem  E,  if  a 
were  of  the  second  kind.  Thus  a  takes  all  numerical 
values  of  the  second  number-class. 

If  we  denote  the  type  of  the  second  number-class 
by  Q,  the  type  of  (4)  is 

w  +  Q  =  w  +  o.)2  +  (C2-w-^). 

[245]    But  since  a)  +  a)^  =  w^,  we  have 

w  +  il  =  il\ 


200     THE  FOUNDING  OF  THE   THEORY 

and  consequently 

o) + n = Q = ^^^. 

G.  If  e  is  any  e-number  and  a  is  any  number  of 
the  first  or  second  number-class  which  is  less  than  e  : 

a<e, 

then  e  satisfies  the  three  equations  : 

a  +  e  =  e,       a^  —  e^      a^  —  e. 

Proof. — If  ao  is  the  degree  of  a,  we  have  ao^a,  and 
consequently,  because  of  a<e,  we  also  have  ao<e. 
But  the  degree  of  e  =  w^  is  e;  thus  a  has  a  less 
degree  than  e.  Consequently,  by  theorem  D  of 
§  19, 

and  thus 

On  the   other  hand,    we  have,    by  formula  (13)  of 

§  19, 

ae  =  ao)"  =  isf-^^^  —  co^  =  e, 
and  thus 

a^e  =  €. 

Finally,    paying    attention    to  the    formula   (16)  of 

§19, 


.H.    If  a  is  any  number  of  the  second  number-class, 
the  equation 


''=i 


has  no  other   roots    than  the  e-numbers  which  are 
greater  than  a. 


OF  TRANSFINITE  NUMBERS        201 
Proof. — Let  /3  be  a  root  of  the  equation 

af  =  |. 
SO  that 

Then,  in  the  first  place,   from  this  formula  follows 
that 

On  the  other  hand,  /3  must  be  of  the  second  kind, 
since,  if  not,  we  would  have 

Thus  we  have,  by  theorem  F  of  §  19, 
and  consequently 

[246]   By  theorem  F  of  §  19,  we  have 
and  thus 

But  /3  cannot  be  greater  than  ao/3  ;  consequently 

ao/3  =  ^, 
and  thus 

Therefore  ^  is  an  e-number  which  is  greater  than  a. 
Halle,  March  1897. 


NOTES 

In  a  sense  the  most  fundamental  advance  made  in 
the  theoretical  arithmetic  of  finite  and  transfinite 
numbers  is  the  purely  logical  definition  of  the 
number-concept.  Whereas  Cantor  (see  pp.  74, 
86,  112  above)  defined  "cardinal  number"  and 
'*  ordinal  type"  as  general  concepts  which  arise  by 
means  of  our  mental  activity,  that  is  to  say,  as 
psychological  entities,  Gottlob  Frege  had,  in  his 
Grundlageti  der  Arithmetik  of  1884,  defined  the 
"  number  {Anzahl)  of  a  class  u  "  as  the  class  of  all 
those  classes  which  are  equivalent  (in  the  sense  of 
PP-  75)  S6  above)  to  u.  Frege  remarked  that  his 
"  numbers  "  are  the  same  as  what  Cantor  (see  pp. 
40,  74,  86  above)  had  called  "powers,"  and  that 
there  was  no  reason  for  restricting  "numbers"  to 
be  finite.  Although  Frege  worked  out,  in  the  first 
volume  (1893)  of  his  Grundgesetze  der  AritJunetik^ 
an  important  part  of  arithmetic,  with  a  logical 
accuracy  previously  unknown  and  for  years  after- 
wards almost  unknown,  his  ideas  did  not  become  at 
all  widely  known  until  Bertrand  Russell,  who  had 
arrived  independently  at  this  logical  definition  of 
"cardinal  number,"  gave  prominence  to  them  in  his 


NOTES  203 

Principles  of  Mathematics  of  1903.*  The  two  chief 
reasons  in  favour  of  this  definition  are  that  it 
avoids,  by  a  construction  of  "numbers"  out  of  the 
fundamental  entities  of  logic,  the  assumption  that 
there  are  certain  new  and  undefined  entities  called 
*' numbers";  and  that  it  allows  us  to  deduce  at 
once  that  the  class  defined  is  not  empty,  so  that 
the  cardinal  number  of  u  "exists"  in  the  sense 
defined  in  logic  :  in  fact,  since  u  is  equivalent  to 
itself,  the  cardinal  number  of  u  has  u  at  least  as  a 
member.  Russell  also  gave  an  analogous  definition 
for  ordinal  types  or  the  more  general  "relation 
numbers. "  f 

An  account  of  much  that  has  been  done  in  the 
theory  of  aggregates  since  1897  ^n^y  t)e  gathered 
from  A.  Schoenflies's  reports  :  Die  Entwickelung 
der  Lehre  von  den  Pmiktmannigfaltigkeiten,  Leipzig, 
1900;  part  ii,  Leipzig,  1908.  A  second  edition 
of  the  first  part  was  published  at  Leipzig  and  Berlin 
in  191 3,  in  collaboration  with  H.  Hahn,  under  the 
title  :  Entwickelung  der  Mengenlehre  und  iJirer 
Anwendungen.  These  three  books  will  be  cited 
by  their  respective  dates  of  publication,  and,  when 
references  to  relevant  contributions  not  mentioned 
in  these  reports  are  made,  full  references  to  the 
original  place  of  publication  will  be  given. 


*  Pp.  519,  111-116.  Cf.  Whitehead,  Amer.  Jotirn.  of  Mai h.,  vol. 
xxiv,  1902,  p.  378.  For  a  more  modern  form  of  the  doctrine,  see 
Whitehead  and  Russell,  Frincipia  Mathetnatica^  vol.  ii,  Cambridge, 
1912,  pp.  4,  13. 

t  Principles,  pp.  262,  321  ;  and  Frtncipta,  vol.  ii,  pp.  330, 
473-510. 


204  NOTES 

Leaving  aside  the  applications  of  the  theory  of 
transfinite  numbers  to  geometry  and  the  theory  of 
functions,  the  most  important  advances  since  1897 
are  as  follows  : 

(i)  The  proof  given  independently  by  Ernst 
Schroder  (1896)  and  Felix  Bernstein  (1898)  of  the 
theorem  B  on  p.  91  above,  without  the  supposition 
that  one  of  the  three  relations  of  magnitude  must 
hold  between  any  two  cardinal  numbers  (1900,  pp. 
16-18;    1913,  pp.  34-41  ;    1908,  pp.   10-12). 

(2)  The  giving  of  exactly  expressed  definitions 
of  arithmetical  operations  with  cardinal  numbers 
and  of  proofs  of  the  laws  of  arithmetic  for  them  by 
A.  N.  Whitehead  {Ainej\  Journ.  of  Math.,  vol. 
xxiv,  1902,  pp.  367-394).  Cf.  Russell,  Principles, 
pp.  1 1 7- 1 20.  A  more  modern  form  is  given  in 
Whitehead  and  Russell's  Principia,  vol.  ii,  pp. 
66-186. 

(3)  Investigations  on  the  question  as  to  whether 
any  aggregate  can  be  brought  into  the  form  of 
a  well-ordered  aggregate.  This  question  Cantor 
{cf.  1900,  p.  49;  191 3,  p.  170;  and  p.  61  above) 
believed  could  be  answered  in  the  affirmative. 
The  postulate  lying  at  the  bottom  of  this  theorem 
was  brought  forward  in  the  most  definite  manner 
by  E.  Zermelo  and  E.  Schmidt  in  1904,  and 
Zermelo  afterwards  gave  this  postulate  the  form  of 
an  ''axiom  of  selection"  (1913,  pp.  16,  170-184; 
1908,  pp.  33-36).  Whitehead  and  Russell  have 
dealt  with  great  precision  with  the  subject  in  their 
Principia,    vol.    i,    Cambridge,    19 10,   pp.    500-568. 


NOTES  205 

It  may  be  remarked  that  Cantor,  in  his  proof  of 
theorem  A  on  p.  105  above,  and  in  that  of  theorem 
C  on  pp.  1 6 1- 1 62  above,*  unconsciously  used  this 
axiom  of  infinite  selection.  Also  G.  H.  Hardy 
in  1903  (1908,  pp.  22-23)  used  this  axiom,  un- 
consciously at  first,  in  a  proof  that  it  is  possible  to 
have  an  aggregate  of  cardinal  number  j^^  in  the 
continuum  of  real  numbers. 

But  there  is  another  and  wholly  different  question 
which  crops  up  in  attempts  at  a  proof  that  any 
aggregate  can  be  well  ordered.  Cesare  Burali-Forti 
had  in  1897  pointed  out  that  the  series  of  all  ordinal 
numbers,  which  is  easily  seen  to  be  well  ordered, 
must  have  the  greatest  of  all  ordinal  numbers  as  its 
type.  Yet  the  type  of  the  above  series  of  ordinal 
numbers  followed  by  its  type  must  be  a  greater 
ordinal  number,  for  /3+  i  is  greater  than  f^.  Burali- 
Forti  concluded  that  we  must  deny  Cantor's  funda- 
mental theorem  in  his  memoir  of  1897.  A  different 
use  of  an  argument  analogous  to  Burali-Forti's  was 
made  by  Philip  E.  B.  Jourdain  in  a  paper  written  in 
1903  and  published  in  1904  (Phitosophical  Magazine, 
6th  series,  vol.  vii,  pp.  61-75).  The  chief  interest 
of  this  paper  is  that  it  contains  a  proof  which  is 
independent  of,  but  practically  identical  with,  that 
discovered  by  Cantor  in    1895,  and  of  which  some 


*  Indeed,  we  have  here  to  prove  that  any  enumerable  aggregate  of 
any  enumerable  aggregates  gives  an  enumerable  aggregate  of  the 
elements  last  referred  to.  To  prove  that  Xo-  Xo  =  Ko,  it  is  not  enough  to 
prove  the  above  theorem  for  particular  aggregates.  And  in  the  general 
case  we  have  to  pick  one  element  out  of  each  of  an  infinity  of  classes, 
no  element  in  each  class  being  distinguished  from  the  others. 


2o6  NOTES 

trace  is  preserved  in  the  passage  on  p.  109  above 
and  in  the  remark  on  the  theorem  A  of  p.  90. 
This  proof  of  Cantor's  and  Jourdain's  consists  of 
two  parts.  In  the  first  part  it  is  established  that 
every  cardinal  number  is  either  an  Aleph  or  is  greater 
than  all  Alephs.  This  part  requires  the  use  of 
Zermelo's  axiom;  and  Jourdain  took  the  "proof" 
of  this  part  of  the  theorem  directly  from  Hardy's 
paper  of  1903  referred  to  above.  Cantor  assumed 
the  result  required,  and  indeed  the  result  seems  very 
plausible. 

The  second  part  of  the  theorem  consists  in  the 
proof  that  the  supposition  that  a  cardinal  number 
is  greater  than  all  Alephs  is  impossible.  By  a  slight 
modification  of  Burali-Forti's  argument,  in  which 
modification  it  is  proved  that  there  cannot  be  a 
greatest  Aleph,  the  conclusion  seems  to  follow  that 
no  cardinal  number  can  be  other  than  an  Aleph, 

The  contradiction  discovered  by  Burali-Forti  is 
the  best  known  to  mathematicians  ;  but  the  simplest 
contradiction  was  discovered  *  by  Russell  {Principles, 
pp.  364-368,  101-107)  from  an  application  to  "the 
cardinal  number  of  all  things  "  of  Cantor's  argument 
of  1892  referred  to  on  pp.  99-100  above.  Russell's 
contradiction  can  be  reduced  to  the  following  :  If 
w  is  the  class  of  all  those  terms  x  such  that  x  is  not 
a  member  of  x,  then,  if  21/  is  a  member  of  w^  it  is 
plain  that  w  is  not  a  member  of  w  ;  while  if  w  is 
not  a  member  of  w,  it  is  equally  plain  that  ze/  is  a 
member  of  w.      The  treatment  and  final  solution  of 

*  This  argument  was  discovered  in  1900  (see  Monisf,  Jan.  1912). 


NOTES  207 

these  paradoxes,  which  concern  the  foundations  of 
logic  and  which  are  closely  allied  to  the  logical 
puzzle  known  as  "the  Epimenides,"  *  has  been 
attempted  unsuccessfully  by  very  many  mathe- 
maticians, f  and  successfully  by  Russell  {cf.  Principles, 
pp.   523-528;  Principia,  vol.  i,  pp.  26-31,  39-90)- 

The  theorem  A  on  p.  105  is  required  (see  theorem 
D  on  p.  108)  in  the  proof  that  the  two  definitions 
of  infinity  coincide.  On  this  point,  cf.  Principles, 
pp.  1 21-123  ;  Principia,  vol.  i,  pp.  569-666;  vol.  ii, 
pp.   187-298. 

(4)  Investigation  of  number-classes  in  general, 
and  the  arithmetic  of  Alephs  by  Jourdain  in  1904 
and  1908,  and  G.  Hessenberg  in  1906  J  (191 3» 
pp.    131-136;   1908,    pp.    13-14)- 

(5)  The  definition,  by  Felix  Hausdorff  in  1904- 
1907,  of  the  product  of  an  infinity  of  ordinal  types 
and  hence  of  exponentiation  by  a  type.  This 
definition  is  analogous  to  Cantor's  definition  of 
exponentiation  for  cardinal  numbers  on  p.  95 
above. §     Cf.    1913,    pp.    75-80;   1908,   pp.   4^-45- 

(6)  Theorems    due    to    J.    Konig    (1904)    on    the 


*  Ephnenides  was  a  Cretan  who  said  that  all  Cretans  were  liars. 
Obviously  if  his  statement  were  true  he  was  a  liar.  The  remark  of  a 
man  who  says,  "  I  am  lying,"  is  even  more  analogous  to  Russell's  w. 

t  Thus  Schoenflies,  in  his  Reports  of  1908  and  1913,  devotes  an 
undue  amount  of  space  to  his  "solution  "  of  the  paradoxes  here  referred 
to.  This  "  solution"  really  consists  in  saying  that  these  paradoxes  do 
not  belong  to  mathematics  but  to  "philosophy."  It  may  be  remarked 
that  Schoenflies  seems  never  to  have  grasped  the  meaning  and  extent  of 
Zermelo's  axiom,  which  Russell  has  called  the  "multiplicative  axiom." 

+  Just  as  in  the  proof  that  the  multiplication  of  Ny  by  itself  gives  Kq, 
the  more  general  theorem  here  considered  involves  the  multiplicative 
axiom. 

§  Cf.  Jourdain,  Mess,  of  Math.  (2),  vol.  xxxvi,  May  1906,  pp.  1 3- 1 6. 


2o8  NOTES 

inequality  of  certain  cardinal  numbers ;  and  the 
independent  generalization  of  these  theorems, 
together  with  one  of  Cantor's  (see  pp.  81-82 
above),  by  Zermelo  and  Jourdain  in  1908  (1908, 
pp.    16-17;   1913,    PP-    65-67). 

(7)  Hausdorff's  contributions  from  1906  to  1908 
to  the  theory  of  linear  ordered  aggregates  (19 13, 
pp.   185-205  ;   1908,  pp.  40-71). 

(8)  The  investigation  of  the  ordinal  types  of 
multiply  ordered  infinite  aggregates  by  F.  Riess 
in   1903,   and  Brouwer  in   191 3  (191 3,   pp.    85-87). 


INDEX 


Abel,  Niels  lienrike,  lO, 
Abelian  functions,  lO,  ii. 
Absolute  infinity,  62,  63. 
Actuality  of  numbers,  67. 
Addition  of  cardinal  numbers,  So, 
91  ff. 
of  ordinal  types,  81,  119  ff. 
of  transfinite  numbers,  63,  66, 
153  ff->  175  ff->  206. 
Adherences,  73. 

Aggregate,  definition   of,  46,  47, 
.54,  74,  85. 
of  bindings,  92. 
of  union,  50,  91. 
Alembert,  Jean  Lerond  d',  4. 
Algebraic  numbers,  38  ff. ,  127. 
Aquinas,  Thomas,  70. 
Aristotle,  55,  70. 

Arithmetic,  foundations    of,   with 
Weierstrass,  12. 
with  Frege  and  Russell,  202, 
203. 
Arzela,  73. 

Associative  law  with  transfinite 
numbers,  92,  93,  119,  121, 
154,  155. 

Baire,  Rene,  ']t,. 
Bendixson,  Ivar,  ']i. 
Berkeley,  George,  55. 
Bernoulli,  Daniel,  4. 
Bernstein,  Felix,  204. 
Bois-Reymond,  Pauldu,  22,34,51. 
Bolzano,  Bernard,    13,  14,  17,  21, 

22,41,  55.  72. 
Borel,  Emile,  73. 
Bouquet,  7. 
Briot,  7. 
Broden,  73. 
Brouwer,  208. 
Burali-Forti,  Cesare,  20s,  206, 


Cantor,  Georg,  v,  vi,  vii,  3,  9,  10, 
13,  18,  22,  24,  25,   26,  28, 

29,  30,  32,  11.  34,  35,  36, 
.      37,  38,  41,  42,  45,  46,  47, 

48,  49,  51,  52,  53,  54,  55, 
56,  57,  59,  60,  62,  63,  64, 
68,  69,  70,  72,  73,  74,  76, 
77,   79,   80,    81,    82,  202, 
204,  205,  206,  208. 
Dedekind  axiom,  30. 
Cardinal  number  {see  also  Power), 
74,  79  ff.,  85  ff.,  202. 
finite,  97  ff. 

smallest  transfinite,  103  ff. 
Cardinal      numbers,       operations 
with,   204. 
series  of  transfinite,  109. 
Cauchy,  Augustin  Louis,  2,  3,  4,  6, 

8,  12,  14,  15,  16,  17,22,  24. 
Class  of  types,  114. 

Closed  aggregates,  T32. 

types,  133. 
Coherences,  73. 
Coherent  series,  129,  130. 
Commutative  law  with  transfinite 
numbers,      66,      92,      93, 
119  ff.,  190  ff. 
Condensation  of  singularities,  3, 

9,  48,  49. 

Connected  aggregates,  72. 

Content  of  aggregates,  ']1. 

Content-less,  51. 

Continuity  of  a  function,  i. 

Continuous     motion     in     discon- 
tinuous space,  37. 

Continuum,  33,  yj ,  41  ff.,  47,  48, 

64,  70  ff.,  96,  205. 
Contradiction,  Russell's,  206,  207. 
Convergence  of  series,   i,   15,  16, 

17,  20,  24. 
Cords,  vibrating,  problem  of,  4. 


209 


14 


2IO 


INDEX 


D'Alembert        {see        Alembert, 

J.  L.  d'). 
Dedekind,    Richard,    vii,  23,  41, 

.  47,  49,  IZ- 
Definition  of  aggregate,  37. 
Democritus,  70. 
De  Morgan,  Augustus,  41. 
Density  in  itself,  132. 
Derivatives    of    point- aggregates, 

3,  3off-.,37. 
Descartes,  Rene,  55. 
Dirichlet,  Peter  Gustav  Lejeune, 

2,  3,  5,  7,  8,  9,  22,  35, 
Discrete  aggregates,  51. 
Distributive   law   with  transfinite 

numbers,  66,  93,  121,  155. 

Enumerability,  32,  38ff.,47,  5ofif., 

62. 
Enumeral,  52,  62. 
Epicurus,  70. 
Epimenides,  207. 
Equivalence    of    aggregates,     40, 

75,  86  ff. 
Euler,  Leonhard,  4,  5,  9,  10. 
Everywhere-dense  aggregates,  33, 

35.  37,  123. 
types,  133. 
Exponentiation       of       transfinite 

numbers,  82,  94  ff,,  207. 

Fontenelle,  118. 

Formalism  in  mathematics,  70,  81. 

Fourier,  Jean   Baptiste  Joseph,  I, 

2,  5,  6,  8,  24. 
Freedom  in  mathematics,  d']  ff. 
Frege,  Gottlob,  23,  70,  202, 
Function,  conception  of,  i. 
Functions,  theory  of  analytic,  2, 

6,    7,    10,   II,   12,   13,  22, 

73. 
arbitrary,  4,  6,  34. 
theory  of  real,  2,  8,  9,  73. 
Fundamental  series,  26,  128  ff. 

Gauss,  Carl  Friedrich,  6,  12,  14. 
Generation,  principles  of,  56,  57. 
Gudermann,  lO. 

Hahn,  H.,  203. 

Haller,  Albrecht  von,  62. 


Hankel,  Hermann,  3,  7,  8,  9,  17, 

49,  70. 
Hardy,  G.  H.,  205,  206, 
Harnack,  Axel,  51,  73. 
Hausdorff,  Felix,  207    208. 
Heine,  H.  E.,  3,  26,  69. 
Helmholtz,  H.  von,  42,  70,  81. 
Hessenberg,  Gerhard,  207. 
Hobbes,  Thomas,  55. 

Imaginaries,  6. 

Induction,  mathematical,  207. 

Infinite,  definition  of,  41,  61,  62. 

Infinitesimals,  64,  81. 

Infinity,  proper  and  improper,  55, 

79-. 

Integrability,       Riemann's      con- 
ditions of,  8. 
Integrable  aggregates,  51. 
Inverse  types,  114. 
Irrational  numbers,  3,  14  ff. ,  26  ff. 
analogy  of  transfinite  numbers 
with,  77  ft". 
Isolated  aggregate,  49. 
point,  30. 

Jacobi,  C.  G.  J.,  10. 
Jordan,  Camille,  73. 
Jourdain,  Philip  E.   B.,  4,  6,  20, 
32,  52,  205,  206,  207,  208. 

Killing,  W.,  118. 

Kind  of  a  point-aggregate,  32. 

Kirchhoff,  G. ,  69. 

Konig,  Julius,  207. 

Kronecker,  L.,  70,  81. 

Kummer,  E.  E. ,  69. 

Lagrange,  J.  L.,  5,  14. 
Leibniz,  G.  W.  von,  55. 
Leucippus,  70. 
Limitation,  principle  of,  60. 
Limiting  element  of  an  aggregate, 

Limit-point,  30. 

Limits  with   transfinite  numbers, 

77  ff.,  131  ff.,  58  ff 
Liouville,  J.,  40. 
Lipschitz,  R.,  6. 
Locke,  J.,  55. 
Lucretius,  70. 


INDEX 


Mach,  Ernst,  69. 
Maximum  of  a  function,  22. 
Mitlag-Leifler,  Gosta,  ii. 
Mutliplication   of    cardinal    num- 
bers, 80,  91  ff. 
of  ordinal  types  81,  119  ff,  154. 
of  transfmite   numbers,  63,  64, 
66,  176  ff. 

Newton,  Sir  Isaac,  15. 
Nominalism,  Cantor's,  69,  70. 
Number-concept,  logical  definition 
of,  202,  203. 

Ordinal        number        {see        also 
Enumeral),  75,  151  ff. 
numbers,  finite,  113,   158,  159. 
type,  75,  79  ff.,  no  ff. 
type    of    aggregate   of  rational 

numbers,  122  ff. ,  202. 
types  of  multiply  ordered  aggre- 
gates, 81,  208. 
Osgood,  W.  F.,  73. 

Peano,  G.,  23. 

Perfect  aggregates,  72,  132. 

types,  133. 
Philosophical   revolution  brought 

about    by    Cantor's    work, 

vi. 
Physical  conceptions  and  modern 

mathematics,  I. 
Point-aggregates,    Cantor's    early 

work  on,  v,  vi. 
theory  of,  3,  20  ff.,  30  ff.,  64, 

73- 
Potential,  theory  of,  7. 
Power,  second,  64  ff. ,  169  ff. 
of    an   aggregate,    32,    37,    40, 

52  ff.,  60,  62. 
Prime  numbers,  transfinite,  64,  66. 
Principal  element  of  an  aggregate, 

131- 
Puiseux,  v.,  7. 

Reducible  aggregates,  71. 
Relation  numbers,  203. 
Riemann,  G.  F.  B.,  3,  7,  8,  9,  10, 
12,  25,  42. 


Riess,  F.,  208. 

Russell,  Bertrand,  20,  23,  53,  202, 
203,  204,  206,  207. 

Schepp,  A.,  117. 
Schmidt,  E.,  204. 
Schoenflies,  A.,  73,  203,  207. 
Schwarz,  8,  12. 

Second      number-class,     cardinal 
number  of,  169  ff. 
epsilon-numbers  of  the,  195  ff. 
exponentiation  in,  178  ff. 
normal    form  of  numbers  of, 

183  ff 
numbers  of,  160  ff. 
Segment    of    a    series,    60,    103, 

141  ff. 
Selections,  204  ff. 
Similarity,  76,  112  ff. 
Species  of  a  point-aggregate,  31. 
Spinoza,  B. ,  55. 
Steiner,  J.,  40. 
Stolz,  O.,  17,  73. 
Subtraction  of  transfinite  numbers, 
66,  155,  156. 

Teubner,  B.  G.,  vii. 

Transfinite    numbers,    4,    32,   36, 

50  ff.,  52  ff 
Trigonometrical  developments,  2, 

3,4,  5,  6,  7,  8,  24ff,  31. 

Unextended  aggregates,  51. 
Upper  limit,  21. 

Veronese,  G.,  117,  118. 

Weierstrass,  Karl,  vi,  vii,  2,  3,  10, 
II,  12,  13,  14,  17,  18,  19, 
20,  21,  22,  23,  24,  26,  30, 
48. 

Well-ordered  aggregates,  60,  61, 
75  ff,  137  ff. 

Well-ordering,  204  ff. 

Whitehead,  A.  N.,  203,  204. 

Zeno,  15. 

Zermelo,  E.,  204,  206,  207,  208. 

Zermelo's  axiom,  204  ff. 


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